I've never used drywall for that, but I wouldn't even try; the intended use
of drywall is to have screws driven through it to grab something "solid," not
into it. Also, it'll probably break getting it into the hole.
The only reason I could think of for using plywood would be because it's a
generally flat surface, thereby making your patch piece lay even with the
surface. But because it's ply, if it's a thin ply, say less than 3/4", I'd
guess it could splinter out on the ends and be destroyed screwing into it
unless you pre-drill it.
I patch a lot of holes, and I use cheapo 1x2x8' #3 pine furring strips cut
about 6" longer than the hole and screw the patch in about 2" past the
border. You might need more than one strip. If there's a stud in the way
going horizonitally, put the strip in vertically. Just make sure there's no
"twist" in the strip so your patch lays flat.
PS - 1 5/8" drywall screws
Plywood or just about any wood-type stuff works just fine for backing behind
drywall. I often put a little yellow glue on whatever I'm using before I
stick it in the hole. Half-inch plywood works great.
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