I may not have all of this correct, but here goes!
GNR = gunner which is the Artillary version of private (pte)
AA would be likely be anti aircraft in the context you have quoted RA is Royal Artillery
The Leicestershire Artillary is likely the Leicestershire Yoemenry, which I think was at that time an artillery Regiment. This likely explains the rank of Lance Corporal, the RA rank would have been Lance Bombardier.
Confusing, yes but a army career well worth filling in the gaps
for further reading, and I have not looked into this, you may find that his Regiment (s) published "year books" (eg The Rifle Bigade Association published The Rifle Brigade Chronicles) If these were published for your fathers Regiment, they can probably be picked up used from something like ABE Books
Hope this helps in a small way