clarah
It appears that the only MIC is for Thomas Murphy 3974 RMF, which you have found. All you can really say is that is a better than evens chance, with the info you havve, that this is him. A Medal Card could be missing or misfiled for example, or peraps he was a Corporal, but made oiur he was a sergent, or was a temp sergeasnt, and that is not on MIC.
This is the index card for the actual medal roll. which should give you a few more details like the battalion he was in.
You can get that by wending your way online through the Nat Arc (Kew) site - their navigation is criminally bad, but if yo go to their shop and find where you can get a quote online, for under 10€ you can get the medal roll entry with the details you have.
You then want to either go to Amazon.com or Abe books or you local library and get a book on whatever battalion f the Munsters he was in, to see where he served.
Also if you are lucky you can get to download a batch of War Diaries. That you find on the online section of Nat Arch. I got a years worth of R Dublin Fusiliers (with 4 other regts I was not interested in) for £3.50 online
The war diaries give a day by day account of the battalion's movements
I can see from medal card that 3974 entered France on 17 Dec 1915, which may give a clue to which battalion he was in, but medal roll should confirm. Also it shows him discharged class Z, look that up on Long,Long, Trail to get exact definition.
Anyway if it is 3974 you have enough to follow his movements.
The battalion history should include the band.
And finally the
Great War Forum which is worth posting on as there are some there that may be able to help yo more than I can