I too had found reference in the Navy List - FLEEMING, Admiral the Hon Charles Elphinstone (1764 - 1840)
Also found he was a promoted Captain in 1794 and later placed in command of the Egyptienne , the ship was badly damaged in an engagement when Fleeming was giving evidence at a Court Marshall of Admiral Calder. Look in 4,800+names for him then click on the page number and it will take you an account of the engagement.
Google books mentions that in 23 July 1805 “as a prelude to Trafalgar” Calder was in an action off northern Spain with a French squadron returning from the West Indies to prevent then joining with another squadron already in Ferrol. He had 15 ships with him, including the Egyptienne, commanded by Fleeming, but the action was indecisive. It was Calder’s failure to engage the combined fleet that led to his Court Marshall in December 1805.
The Egyptienne was captured from the French in the West Indies in May 1804. The sea war with the French was far and wide, so you should consider the possibility Henry Rae was captured in the West Indies or the Mediterranean not just in a raid of the French coast. As he claims to have served under Fleeming, as this must have before his promotion to Admiral.
July 1802, Fleeming, Captain in RN, elected MP for Stirling.
Sept 1804, notice is given for Officers & Company of HMS Diomede [Fleeming, Commander] who were actually onboard at the capture of: El Buen F[or S] ardine, 18 October 1800; Chesterfield, 30 November 1800; NuestraSenhora de Monserrat, 3 December 1800. That proceede from the captures will be made at …on Monday 1 October 1804 and everyday for 3 years. Seamen’s share given as £3 15s 11p [£3.79p]
May 1805, notice is given that proceeds from the sale of prize ships and cargo, captured in action of 13 Feb, by L’Egyptienne, Capt. CE Fleeming, will be delivered to Admiralty’s registry.
May 1805, notice given that Officers and Company of L’Egyptienne [Fleeming, Captain] who were on board at capture of El Dichoso, F. Caselins [&] Master, on 13 Feb 1805, will be paid respective portions of full share on board Thur 29 May all unclaimed shares to be paid at …. For3 years.
Dec 1806, notice given…. L’Egyptienne [Fleeming, Captain] …capture of L’Etion, 2 October 1805, ….etc. Mar 1808, notice given ….L’Egyptienne [Fleeming, Captain] … capture of L’Edouard, 1 Nov 1805 ….etc.
July 1821 to Jan 1837 - lists Fleemimg’s promotions [already an Admiral] to various Admiral’s ranks and positions in the Navy.
Depending on when he was captured, it looks like Henry Rae may have missed out some considerable [in those days] prize money.
Difficult to find mention of Fleeming's 1st commission in the RN or subsequent promotions in the Gazette but I've not checked the Times. If that fails to beef up his career or any other sources your aware of, then if your still interested there is a free search, pay for reports, site called the Naval Biographical Database. I’ve never actually used them for any reports but Fleeming is listed as Fleming, unfortunately with 22 records the report will cost £15 UK by email or + p&p for snailmail.
I can confirm the officer Henry Wraye was not your Henry. There’s a document in ADM 6/349/37 at the National Archives entitled:
ff. 165-170. Catherine B Raye, widow of Henry Raye, Lieutenant Royal Navy who died 18 Oct 1812. Includes: Extract from Parish Register, married 15 May 1795. Papers submitted to the Charity for the relief of Officers' Widows.
I think that’s too much of coincidence for it not to be him, especially as I believe [nl1812] = navy list of 1812, the year he died. He’s also listed in the Naval BD with date died as 1812.
That's about as much I can do. Good luck with your research.