I couldn't find a Henry Thomas Caines. I did, however, find Henry Thomas Cains, birth recorded in the first quarter of 1871, in the Keynsham registration district. Couldn't make out on his attestation form exactly where he was born, but it says "near Bristol."
Below is an early list of the section commanded by Lieutenant G. A. Gibbs, presumably the same section that Henry Caines served in, except that he's not listed (he's not named in the other three sections either). There was a W. F. Caines though. Did he ever see active service?, or maybe the initials "W. F." are incorrect.
"The following is a complete list of the members of the North Somerset Company. Following each name is the place where reside the Yeoman's next-of-kin, and which in almost every case is also his home."
......No. 1 Section.
......Officer: Captain G A Gibbs
......Sergeant A Seeley, Castle Cary
......Corporal A Travis, Worle, near Weston-super-Mare
.............."........A Yendole, Nailsea
......Private E Bennett, Bristol
............."......E Blackmore, Ashley Down, Bristol
............."......W F Caines, Pont-street, London
............."......H G Denham, Blackford, near Weston-s-M.
............."......F Dowding, Mitcham
............."......H Dowding, Wincanton
............."......W J Ford, Milton, Weston-super-Mare
............."......W Haggett, Pilton, near Shepton Mallett
............."......F Hamilton, Port Elizabeth
............."......T S Hill, Barnstaple
............."......N C Howes
............."......T Hicks, Hewish, near Weston-super-Mare
............."......J J Kiddell, Tickenham, Yatton
............."......G King, Berkeley
............."......H R Luckman, Weston-super-Mare
............."......F Medlam, Evercreech
............."......H Millard, Theale, Wedmore
............."......F C Ovens, Evercreech
............."......F G Patch, Barrow Gurney
............."......P Seeley, Westbury-sub-Mendip
............."......W H Singer, South Brewham
............."......G Sperring, Cheddar
............."......E Sperring, Cheddar
............."......J Swanton, Gillingham
............."......J D Tabor, North Barrow
............."......A Winstone, Wraxall (orderly to Major Sherston)
The Weston Mercury, February 1900
Possibly William Frank Caines, birth recorded in the 4th quarter of 1864 in the Dorchester registration district.
Or William Frederick Cains, 1st quarter 1877 at Kensington, London.
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* Should anyone want to see the names and places of the men in the other three sections, I have them.