How can I patch a hole in a painted wood paneling (or plywood wall)?
The process for patching should similar to drywall, but with slightly different materials. I'd suggest using wood filler at the joint.

In more detail: first, get a new piece of plywood of the same thickness as the existing wall board. Then you need to attach your patch piece to structure. For this, you have two options.


Cut away more of the existing plywood until you've exposed some structure you can fasten into.
Cut some lumber to span the opening, and attach it using screws through existing plywood around the opening. Then cut your patch and mount it to the spanning pieces.


If the hole is only 6" wide, a small piece of 2x4 or even 1x2 and a plywood patch will probably cover you.

To seal the seams between your patch and existing plywood, use wood filler. Fill with a putty knife, sand flat, prime and paint. Assuming you can find a 4ft x 2ft plywood panel of the appropriate thickness, total materials cost should be around $20-$30.