Thanks to the letter from the PRO in 1950 to Kathleen May O'Hara (b 1907) (via Derek O'Hara), we managed to track down the letters relating to the sale of his commission in WO31/509. Below is the transcription and to the right are links to PDF's of the scans of the original letters.
Letter 1 (Brough offers money for purchase) Buckingham St, Strand 12th January 1821 Sir We beg leave to report, for the information of His Royal Highness the Commander in Chief, that the friends of Lieutenant R Brough of the 56th Regiment have lodged in our hands the sum of eleven hundred pounds as the regulated difference between the value of a Lieutenancy & Company We have the honor to be, with respect Sir Your most Obedient Humble Servant Mssrs Donaldson and Co.
Letter 2 (Placing Lawrence on Half Pay) Horse Guards 25th June 1821 Sir I have the Honor to refer to you the annexed copy of a Medical Report dated 15th instant upon the Case of Bt Major O'Hara of the 56th Regiment who has been absent from his Regiment in consequence of Ill Health since the 5th June 1819, and to acquaint you, that he has been informed in reply to an Application for Four Months further leave of Absence, that as no expectation is held out of his efficiency, the Commander in Chief cannot comply with his request, and he must therefore be placed upon the Half Pay I have the honor to be, Sir Your most Obedient Humble Servant
Letter 2 - Annex (Lawrence's Medical report) Limerick 15th June 1821 Copy I hereby certify I have this day carefully examined Bt Major O'Hara of the 56th Regiment and find that his Health has been for the last three years much impaired by repeated Attacks of Rheumatic Gout & a liver Complaint attended with a [tropical?] affliction of the lumbar. His General Constitution, and great Debility, which he reports was caused by Thirty Years Service in the East Indies. I am of opinion that Bt Major O'Hara is at this time unable to proceed to his Regiment, and further declare my belief that it does not appear to me there is any probability of this Officer becoming again fit for the Active Duties of a Military Life. signed P Ormsby
Letter 3 (Covering Letter requesting leave to sell Commission) Limerick 12th December 1821 Sir I beg leave to produce a Memorial to His Royal Majesty the Commander in Chief for leave to sell my Commission as Captain in His Majesty's 56th Regiment of Foot and which you will be pleased to present. I have the honor to be Sir Your most Obedient Humble servant L O'Hara Bt Major HM 56th Regiment
Letter 4 (Letter requesting leave to sell Commission) Limerick 12th December 1821 To His Royal Majesty the Commander in Chief May it please your Royal Highness to obtain for me His Majesty's consent to the sale of my commission of Captain in His Majesty's 56th Regiment of Foot. In case His Majesty will be graciously pleased to permit the same, I do declare and certify upon the word & honor of an officer and Gentleman, that I have not demanded nor accepted, neither will I demand nor accept directly or indirectly at any time or in any manner whatever, more than the sum of Eighteen hundred pounds being the price limited & fixed by His Majesty's Regulations as the full value of said Commission
Letter 5 - Purchase Money transferred to Agent by Brough 42 Great Portland Street 7 January 1822 Sir I have to thank you for your communication of the 3rd inst. and in reply have the honor to inform you that Mssrs Donaldson and Co. the agents of the 56th Regiment are prepared to pay the purchase money for Lieut. Richmond William Brough when called upon. I have the honor to be Sir Your most Obedient and very Humble servant William Brough