TSB,
A little more information regarding James Crompton from The National Roll of The Great War :-
"CROMPTON, J., PRIVATE, 20TH MANCHESTER REGT.
He volunteered in 1915, and later in the same year was drafted to the Western Front where he saw much severe fighting at St Eloi , Ypres, Festubert, Hill 60, Albert, Armentieres, Arras and Vimy Ridge. He made the ultimate sacrifice during the Somme offensive in July 1916 and was entitled to the 1914-15 star, general service victory medals.
4 Catherine St, Manchester."
This throws up a couple of discrepancies.
The date of death you have varies from this account though I would be more inclined to believe the one you state.
As BumbleB correctly states, his medal roll index card shows an entitlement to the General Service Medal and Victory Medal but no mention of the 14-15 star. The 20th Manchesters (5th Manchester Pals) first embarked for France via Folkestone on the 9th November 1915 which would have entitled him to the Star. The only thing that immediately springs to mind is that perhaps he was injured during training or sick on the date of departure and only re-joined his Battalion in France after the 14-15 Star qualification date of 31st December 1915. The medical pages of his service record would have confirmed or dispelled this theory but I can't find his record so I assume his was one of the many destroyed during the Blitz.
If you wish to research deeper the Manchester Regiment Archive is held at Tameside Local Studies Library and the Regimental Museum is also in Ashton under Lyne. I can thoroughly recommend Michael Stedmans well researched book "The Manchester Pals" which includes details of the 20ths exploits from their very beginning in 1914 and will furnish you with details of where he would have engaged in battle. The 20ths war diary would give you a more day to day account of his service and though it is unlikely to mention him by name it would indicate where he was killed. The diary is available to download from The National Archive for about £3.50.
There is a specific website for The Manchesters and there are many helpful experts who I'm sure would be happy to help you.
Hope the above is of use to you and good luck with your search
Regards
Simon