The permutations could be "A H Greenway-Hunt" "A H Hunt-Greenway" "A H H Greenway" A Hunt Greenway" etc. Nothing in TNA for either , a Dragoon with a MIC. I think that he has probably added the Greenway (or the Hunt) out of nowhere. There are a number of "A H Hunt" but nothing I can find with Greenway. It is written as "Hunt-Greenway Lt AH" in the Register and the other way round in Leave Book. And another rendering has him as "A G Hunt"
I think the clue is in his parents marriage in Odessa in 1869. Name: Henry William Hunt Birth Date: 1841 Age: 28 Spouse's Name: Nina Annibal Spouse's Birth Date: 1851 Spouse's Age: 18 Event Date: 15 Nov 1869 Event Place: Odessa, Kherson, Russia Father's Name: John Greenway Hunt Spouse's Father's Name: John Annibal. So Greenway was his grandfather's middle name. His father was a Consular Assistant when he married
1883 Oct 29. Born Odessa, Russia son of Henry William and Nina Anabel Hunt - ties in with his brother as next of kin. Birthplace: Odessa,
1901 census A pupil at a school in Jersey
1915 Jan 19 Enlisted aged 32 An Engineer of 35 Gloucester Rd, Regents Park, London. Next of kin was William Henry Hunt living in Russia
1916 Sep 6. Cadet Alfred Henry Hunt to be temp. 2nd Lt. (on prob.). Seems Correct. He was in Dragoons, but there is no MIC
1920 Sep 28 . Cl. GG.—Temp. Lt. A. H. Hunt, Gen. List
.
1920 Sep 29. Relinquishes his commission. Temp. Lt and retain the rank of Lt.: — A. H. Hunt. 29th Sept. 1920
1921 Apr 14. Joins ADRIC with service no 1899. Posted to R Coy
Fined £1 by Coy Commander
1922 Jan 17. Discharged on demobilisation of ADRIC
1930 Dec 23. Died (on service record) It could be the man died Epping, reg Jan/Mar 1931, but who was 47 rather than 52 in BMD
WO 339/61336 Alfred Henry Hunt