The following records in the National Archives and the Navy List
indicate that there may have been more than one Royal Navy Surgeon by the name of Anthony Donoghoe........
Name Donoghoe, Anthony Rank: Surgeon Date of Appointment: 22 January 1814 Date: 22 January 1814 Held by: The National Archives
, Kew. Reference: ADM 196/68/5
Name Donoghoe, Anthony Rank: Surgeon Date of Appointment: 17 July 1816 Date: 17 July 1816 The National Archives
, Kew. Reference: ADM 196/8/86
Assistant surgeons Anthony Donoghoe and James Osborne were appointed to the Glasgow
in July 1821 
Anthony Donoghoe is included in the Navy List
of Surgeons date of appointment 6 September 1825
Anthony Donoghoe was employed as surgeon superintendent on the convict ships City of Edinburgh
in 1832, Parmelia
in 1834, Hive
in 1835 and Calcutta
in 1837 all to New South Wales.
City of Edinburgh
1832 - no deaths
This was Anthony Donoghoe's first voyage as Surgeon Superintendent of a convict ship. He kept a Medical Journal from 2 February to 11 July 1832.......Although the medical cases had been few there were a number of prisoners who made vague complaints of pain around the heart which he found on examination was pain around the abdomen arising from indigestion because of sea sickness. As well as illness, two or three cases of general debility, old age and the cases of the women and children belonging to the guard caused him to expend the supply of medical comforts by the time they had reached Sydney. 
1834 - 2 deaths
Medical Journal from 18 September 1833 to 20 March 1834. His first case on the sick list was dated 5th October 1833 when he treated Major Joseph Anderson who was suffering with rheumatism.
1835 - 1 death
Medical Journal from 12 July to 10 December 1835. There was only case of death which happened on board was that of a boy whose early habit of life predisposed him to pulmonary disease contacted during his apprenticeship to a glass manufactory. On the Night of Thursday 10th December 1835, the Hive ran aground on a sandy beach south of Jervis Bay and was wrecked.
1837 - 9 deaths
Medical Journal from 25 March to 4 August 1837....... His first case on 19th April 1837 was that of a soldier of the 80th regiment who had been unwell for 10 or 12 days and was no longer able to do his duty. He had become ill on the journey from Chatham and had been further exposed to the cold in Kingstown harbour when on the open deck of the steam vessel conveying the military guard to the Calcutta. In the early part of the voyage the prisoners suffered mostly minor ailments, however many had been used to a diet that consisted of potato and from the sudden change on board ship and in conjunction with sea sickness many suffered from obstinate constipation. By mid July, three months into the journey, sixteen of the men were affected by scurvy.
In the 1841 Census Anthony Donoghoe, surgeon, is recorded as age 49 and residence at Clarendon Square St. Pancras. 
In 1842-43 Anthony Donoghoe was appointed to the Dido
in the East Indies 
Anthony Donohoe died on the 16 October 1847 in Killeshandra, Co. Cavan, Ireland 
 Navy List
 NAVAL PROMOTIONS and APPOINTMENTS . Glasgow Herald (Glasgow, Scotland), Monday, July 9, 1821; Issue 1935. British Library Newspapers, Part I: 1800-1900.
 THE DUBLIN MARKETS . Freeman's Journal and Daily Commercial Advertiser (Dublin, Ireland), Wednesday, October 20, 1847; Issue N/A. British Library Newspapers, Part I: 1800-1900.
 The Navy List
 Ancestry.com. 1841 Census. Class: HO107; Piece: 685; Book: 14; Civil Parish: St Pancras; County: Middlesex; Enumeration District: 14b; Folio: 5; Page: 5; Line: 1; GSU roll: 438801
 Journal of Anthony Donoghue. Ancestry.com. UK, Royal Navy Medical Journals, 1817-1857. Original data: The National Archives. Kew, Richmond, Surrey.