Susie,
The LSGC appeared in regimental orders and on a mans papers, although the MSM was gazetted, it would as be worth your while taking a look in WO23 and WO101 for Clarke, in fact from memory, all the immediate ones from, I think, 1915 to about 1928 were also actually published in a book with the actual London Gazette reference and are searchable by name, but, you then would still have to look at the relevant gazette for the particular mans regiment/corps.
So that is one way to go, another is, given your second photograph, which very clearly shows a WO1 in uniform, would be to look at every Frederick/Fredrick John Clarke, every Frederick Clarke and every Frederick J Clark in WO372, as in the second photograph he wears the ribbon for the Star, we can assume the photo is circa 1917 or later, the index cards in WO372 will normally show the highest rank held, whilst actually employed in the service, that would certainly narrow the field, lots of men with that name, but, not many with that particular rank.
Then all that is needed is to take each particular man you have found, look at WO329 for any other information relating to his Great War medals which may not appear in WO372 and search for any surviving papers in WO363 and WO364, but, you should note that a manual search in WO363 is not possible for obvious reasons, a great deal was lost when the Army Records Office was damaged during the blitz.
Now, regarding this latter, I had already looked at the available folio and nobody jumped out at me with this name, but, I based that on the original information that you gave regarding birth details and earlier service in the Suffolk Regiment.
You should also note that some have not been indexed properly and you can, on occasion, find yourself looking at a particular man and upon looking at the next folio, you find yourself looking at a completely different man.
Anyway, easier by far, to just talk to the family and find those medals, not easy things to just loose given they are mounted for wear, but, of course, very easy to pawn or have sold.
Regards Frank