I would suggest it's "COMPy" [Company] A. W.G. [W..?.. Guards], designating some Militia company beginning with a 'W'.
Some years ago I posted on CWT a link to a 19th century book that listed all the company/battalion/regimental names or nicknames used by the Union army or state militias. That source would quickly identify all "W.G." units, at least on the Federal side. I have not been able to locate that post, or the book it links to. Perhaps someone remembers it?
I agree it seems like it's William H. Hood Company A W.G. (Wish the name was a bit more unique)
Here is the book of unit synonyms:
https://archive.org/details/listofsynonymsof00volurich/page/296
I came up with 10 potential Union W.G. synonyms (other Co. letter designations excluded):
(*) Means no William Hood on the register/muster roll for that unit.
*Wadsworth Guards NY 104th Regt. Infantry
*Wayne Guards, Anthony NY 135th Regt. Infantry (6th Regt. Heavy Artillery)
*Washington Greys NY 8th Regt. Militia Infantry (National Guard)
*Washington Greys NY 47th Regt. Infantry
*Washington Greys NY 1st Regt. Engineers (Artillery Co. L) 4th Ind. Battery
Wallace Guards OH Charles Worthington's Independent Company (30 days)
Washington Guards PA 60th Regt. Militia (90 days)
Watson Guards WI 11th Regt. Infantry Co. A
Wayne Guards PA 12th Regt. Reserves Co. A
Wormer Guards MI Wormer Guards Co. A&B (1864)
There was a William H. Hood in the Washington Grays (Co. F 17th PA Infantry - 90 days)
Favorite one I saw: What Cheer Guards from RI