The Horse Guards ( also called The Blues) were a cavalry regiment that, together with the Life Guards, were part of the British sovereign's Household Brigade. Back in 1791 I think they were barracked at Hyde Park then moved to Windsor, their headquarters now is in Whitehall, which is also the headquarters of the British Army.
As Far as MAYNE goes
I have Ens. Charles MAYNE joined the 43rd Foot on 25 Nov. 1808 and died in 1813.
Lt.Richard MAYNE was in the 30th Foot died in 1826 also served in Netherlands 1814
Lt. William Mayne of the 59th Foot Joined 4 September 1808 Served at Waterloo
**Lt. William MAYNE was in the Life Guards joined up on 26 September 1811 Served at Waterloo and he died 0n 11 November 1843

To be the seargent MAYNE at Richard Palmer's trial Lt. William MAYNE may have been a seargent in 2nd troop of Horse Guards before joining the Life Guards in 1811

Just noticed I have two Lt Mayne's up there (I didn't space) disregard the William from the 59th Regt of Foot. They had nothing to do with the household guards. Look to the last one in the Life Guards.
In the meantime I'll see if I can find your Richard Palmer in the Horse Guards in 1791 that would probably have been under Col.Jeffrey Amherst.