Richard Clarke's appointment as Surgeon to the Bellona in 1792 was noted in correspondence to Governor Phillip dated July 1792 (Extract) -
Commissioners of The Navy to Governor Phillip
Navy Office, 25 July 1792
The Lords Commissioners of the Treasury having thought fit that a skilful surgeon should go out in the Bellona to have the care of the settlers and convicts, and also the charge of Government stores on board her, we have appointed Mr. Richard Clarke, a surgeon in the Navy, to perform those services, and desire when that is done you will order him a passage home by the most speedy conveyance that offers. etc
Return to England
Richard Clarke returned to England on the Kitty in June 1793. Surgeons Denis Considen and Alexander Purvis Cranston also sailed on the Kitty 
Richard Clarke married Janet Borwick at St. Mary's Portsea, Hampshire on 28 April 1798.
There are two publications mentioned in records in the name of Richard Clarke R.N.,
1). Description from the Monthly Review in 1795....Plans for increasing the Naval Force of Great Britain, by rendering the Service a more desirable Object to Officers and Seamen, - By Richard Clarke, M. D. Surgeon in the Royal Navy.
Dr. Clarke's suggestions appear to merit the consideration of the minister, and of the Admiralty Board.
His first object is the present rank of Master and Commander, the several inconveniencies attending which distinction from a Post Captain he points out, and which he would remedy by incorporating the two ranks; making a Captain rank as Major for the first three years, as Lieutenant-Colonel for the second three years, and then as Colonel —
The second object is the propriety of introducing into the navy the rank of Second Lieutenants - as in the Marines; in order to prevent the risk that a young man at present incurs, after having served his time and passed his examination as Midshipman, of not being employed as Lieutenant.
The third plan proposed by Dr. C respects the Master's Mates; and he would have experienced men in the merchant's service encouraged, by sufficient wages, to enter the navy as Master's Mates, and to rise to become Masters:- due examinations having pasted each time at the Trinity House.
The last subject here discussed is the mode of inducing able seamen to enter voluntarily into the navy; and this principal objection to serving in the navy, as supposed by Dr. C. to operate among sailors. Being the frequency and the readiness with which corporal punishment is inflicted in ships of war, he proposes that punishment should only be imposed by the judgment of a stated number of officers; and he suggests some regulations respecting disrating and turning before the mast. Some other less material alterations are proposed, and much good sense and reasoning support the author's ideas; of which we have given only a brief outline. 
2). Medical strictures, an effectual method of treating most diseases, in which the prevention, palliation and cure are pointed out. - 1799 ...The Doctor has infallible remedies for colds, small pox, measles, worms and in short for every disease afflicting the human body; differing in this from many of his brethren, that he varies his medicines he says, according to the nature of the diseases; but they are all equally infallible. Those who are desirous of experiencing the utility of them are desired to apply to the Doctor in Gough Square or to any or all the booksellers in Great Britain. - British Catalogue. See Internet Archive for the full publication.
Navy List 1814
Richard Clarke was included in the Navy List of Medical Officer in 1814. (Retired)
Notes and Links
1). Navy pension: memorial of Richard Clarke, surgeon Date: 1799 July. Reference: PC 1/44/156 - National Archives
2). Application for Charity - Janet Clarke, widow of Richard Clarke, surgeon Royal Navy who died 24 Oct 1818. Includes: Extract from Parish Register, married 28 Apr 1798. Papers submitted to the Charity for the relief of Officers' Widows. Date: 1818. Reference: ADM 6/356/61 Description: ff. 330-334. National Archives