The City of Edinburgh was the next convict ship to leave Ireland bound for New South Wales after the departure of the Mangles in February 1828 and the next female convict ship to leave Ireland after the Elizabeth in August 1827. 
The City of Edinburgh departed Cork on 23rd June 1828 with 80 female prisoners, 12 free women + 36 children
Mrs. Sarah Clendon and child, wife of the Captain arrived as passengers.
Surgeon William Anderson
William Anderson kept a Medical Journal from 14 June to 27 November 1828...... He remarked in his journal of the difficulties of mothers nursing babies. - There were six children at their mother's breast during this voyage and these women appeared to suffer more from debility than the others their provisions being scanty and the children not having any allowance. I endeavoured to remedy this by allowing them some of the medical comforts in my charge. 
The free children treated by surgeon Anderson included -
Edward and John Gilligan both aged 13;
Nicholas Gilligan aged 14;
Walter Birmingham aged 12;
Patrick Birmingham aged 9;
Betty Birmingham aged 8;
Bridget Lynch aged 12;
James Grady age 13;
Thomas Shanahan aged 12;
Ann McGuigan aged 14;
Ellen Grady aged 16.
As well as the free children there were some very young convict girls on the voyage -
Catherine Byrnes aged 16;
Honora Crotty aged 17;
Honora Harrington aged 16;
Mary McAndrew aged 16;
Abbey Murphy age 14;
Margaret O'Donnell aged 15.
William Anderson had to manage without the usual list of convicts. He later expressed his frustration in correspondence to Colonial Secretary Macleay .......
Ship City of Edinburgh,
Sydney Cove 10th December 1828
In reply to your letter of the 8th inst., received this morning containing directions from His Excellency the Governor to me to transmit you the list of the convicts, who came out in this ship with their characters prior to embarkation, supposed to have been received by me agreeably to the directions of the Home Secretary of State I have to state to you for His Excellency's information that I am not in possession of any document of the kind. I applied personally to Doctor Trevor, the Gentleman who superintended their embarkation, who refused to furnish me with it; I have had much reason to regret this refusal, having been obliged to grope my way among them perfectly unacquainted with their previous characters and habits and even of the crimes of which they were convicted. (Historical Records of Australia, Vol. XIV, p. 565)
The ship was kept clean by having the beds on deck every day and frequently opened to be aired. The Berths deck was well cleaned every day and kept as dry as the state of the weather would admit and swept up after every meal; the bed places were swept out every day and scrubbed twice a week; the lower bed berths being raised to clean the deck beneath them; the airing stove was frequently alight according to the state of the weather and the swinging stove almost every day in the Hospital or other parts of the deck where most required. The women had perfect liberty at all times to come on deck from eight o'clock in the morning to sunset and this no doubt had its effect in keeping them in good health their spirits becoming now buoyant and the depressing passions which strict confinement might have engendered avoided. When they were landed a fortnight after our arrival there was not a patient on the sick list.
Arrival at Port Jackson
The City of Edinburgh arrived on 12th November 1828, a voyage of 142 days.
One more person was landed than embarked, a child being born on the passage out (Catherine Ahern gave birth to a child on 14th September).
A Muster was held on board by Colonial Secretary Alexander Macleay on 14th November 1828. The indents include name, age, education, religion, marital status, family, native place, offence, when and where tried, sentence, prior convictions, physical description and where the women were assigned on arrival. There are also occasional notes about the women:
Sady Allen was the wife of Robert Read a boot and shoemaker in America. She died in Sydney Hospital 21 April 1832
Judy Allen - Nose cocked showing cartilage
Catherine Ahern was the wife of Richard Short a victualler in Ireland
Catherine Burns died at Parramatta in July 1833
Honora Crotty - Well looking. Nurse girl Age 17
Margaret Farey age 50 - bad countenance
Catherine Fitzgerald - Husband John Fitzgerald now a confection cook in Galway
Margaret Grady - full mouthed
Margaret Hayes - Good countenance
Mary Heffernan - Husband Mr. John B. Freeman, Magistrate Co. Cork - Note Ridiculous, idiotic
Catherine Hughes - Well looking. Husband John Hughes in 1st Battalion of Royals.
Elizabeth Johnstone - Husband in the colony as Owen Connor per Borodino
Mary Lane - Age 22. Husband Joseph Lenehan a soldier in 28th regiment. Says she was born with grey hair
Mary Long - One son Richard Long here per Borodino, the other in Bandon Co. Cork
Bridget Lennon - Husband in colony as Andrew Lennon per Mangles 5
Margaret Mahoney - Husband in colony, John Mahoney per Borodino
Bridget Murphy - One daughter on the colony, Bridget Murphy per the Elizabeth
Margaret Mara - Husband John Mara a farmer in Tipperary
Ellen McCarthy or Joyce - Husband John Joyce a pedlar in Limerick
Bridget Neill - Husband W. Cusack a labourer in Limerick
Mary O'Brien - Husband in colony as ?Lance O'Brien. Was here formerly as a free servant. Went home in the Midas 4 years since
Catherine Quinn - Husband Jeremiah Keefe a stonemason. Subdued cast of countenance
Ellen Regan - well looking
Catherine Ruane - sickly with one child
Ellen Sullivan - Husband John Rebsey a blacksmith in Cork
Ann West - Husband convicted and expected to arrive. Michael Forbes
As can be seen in the Report below, female servants were in an abundance at this time. The Australian advised that:
The Female Prisoners, who arrived by the ship City of Edinburgh, are not off hands yet. Persons of family wanting to have servants, are desired by a Government advertisement, which appears in the back page to make application, in strict conformity with established form before this 22d instant 
The women who had children with them were taken the Female Factory at Parramatta. The other women were assigned to various settlers or to Sydney town where many were assigned to free women.
Management of Parramatta Female Factory 1828
The conduct of the women during the half year has been good, with a few exceptions; some two or three of the most troublesome, having been obliged to be punished as often as six times : punishment has, however, decreased notwithstanding the increased number in the establishment. But 261 cases of punishment occurred during the half year, showing a diminution of 59 cases, on that of the preceding, proving most satisfactorily the advantages arising from the improved order and discipline of the establishment. The crimes are of an ordinary character, and therefore do not appear to call for any particular observation from the Board, further than was noted in their last Report, being of the same character; viz. drunkenness, insolence and obscene language.
The Board find the regulations of the establishment duly observed: the books required by the establishment are in good order, and are neatly kept by the parties having charge of them.
The Matron states, that morning and evening prayers are read by herself to the Protestants and by one of the monitresses to the Catholics; and that good order and attention is observed. It is pleasing to the Board to have it in their power to report their continued satisfaction at the cleanliness, order and regularity observable in the establishment: and their approbation of the manner in which the matron, Mrs. Gordon, performs the arduous duties of her office. The assistant monitresses of the different classes, who are selected from among the prisoners of the first Class, give satisfaction in their appointments. The employment of the women of the establishment, is attended to as far as possible.
The result of each half year, proving that the establishment is increasing, owing to the importations exceeding the demand for female labour, it is with feelings of some satisfaction, the attention of the Board has been directed to the subject of a modification of the present system of granting Tickets of Leave and Indulgences to the more meritorious of this class of females, as called for by the Colonial Secretary's letter. The Board are as much disposed to believe as to hope, a measure of this nature may be attended with beneficial results. In considering the subject, they have to propose their opinion, that Tickets of Leave might be granted as follows; viz.
To women under sentence for Seven years, for continued good conduct in service, or in the married state....2 years
To women under sentence for Fourteen years, for continued good conduct in service, or in the married state.... 3 Years
To women under sentence for Life, for continued good conduct in service, or in the married state - 4 years
As the first Class of this establishment is considered an asylum, and not a place of punishment, for the unappropriated female prisoners; women returned to the first Class will not be considered as having forfeited their claim to a Ticket of Leave, as it must be inferred they have committed no offence in their service, or they would have been punished by being sent to the third Class.
The same indulgence might also be extended to Monitresses in the establishment, who retain their situations for the periods specified, such employment being considered as a service.
Women who have been permitted to marry, should their husbands die or absent themselves, if they have lived two, three or four years, according to the period of their sentence, without being complained of, or becoming known to the police, to be equally eligible to have Tickets of Leave.
In reviewing generally the State of the Establishment, its operation and progress during the half year, the Board are pleased to have it in their power to report favourably on all its branches, with the exception of the increase to its numbers, over which, arising as it has done, from the extent of the importations, they can have no control: Punishments have diminished, and although the labour of the women has not been pushed so far as could be wished, its value has increased sufficient to diminish the expense of maintenance below what it was calculated at the last half year.
In concluding this Report, the Board have much pleasure in adverting to the circumstance of the formation and introduction into the establishment of a Ladies Committee, composed of the principal ladies in the neighbourhood, actively encouraged and patronized by Mrs. Darling; the object of which is, by the adoption of an evening school to inculcate moral instruction, and excite and raise into being a better feeling for their condition, and a desire to improve (by becoming good servants) their stations in society. The ladies also hold out encouragement to work during their leisure time, intending from the proceeds of their labour, to form a Fund from which to reward those who have remained long in their services, with good characters, etc. The Board need hardly remark, that they anticipate much good to result from the interest taken in the establishment by the Ladies Committee.' Female Factory Paramatta, eighth June 1829. -
Extract of a Report of the Board of Management of the Parramatta Female Factory for the year ending December 1828
Kerin, Honora Neill, Bridget
Notes and Links
1). Free passengers arriving on the City of Edinburgh included: Mary Birmingham, Mary Ann Burnett, Mary Elliott, Bridget Gilligan, Mary Grady, Mary Hogan, Mary Lynch, Jane McGuigan, Rose Murray, Johanna O'Neil, Catherine Shanaghan and Mary White. (State Records NSW shipping List)
2). The City of Edinburgh was one of three convict ships bringing female prisoners to New South Wales in 1828, the others being the Elizabeth and the Competitor. A total of 471 female convicts arrived in the colony in the year 1828.
5).National Archives - Reference: ADM 101/17/3A Description: Medical journal of the female convict ship City of Edinburgh for 14 June to 27 November 1828 by William Anderson, during which time the said ship was transporting female convicts to New South Wales
 Ancestry.com. UK, Royal Navy Medical Journals, 1817-1857 . Medical Journal of William Anderson on the voyage of the City of Edinburgh in 1828. The National Archives. Kew, Richmond, Surrey.
 Bateson, Charles Library of Australian History (1983). The convict ships, 1787-1868 (Australian ed). Library of Australian History, Sydney : pp.348-349, 386