How to repair a hole thru linoleum and plywood?

Only need 1/4 underlayment for vinyl .If you have another piece of vinyl you can cut a square out of it.Trace cut it out.fill the plywood with floor patch.let it dry completely
glue the new piece in with a vinyl trowel.Get a seam sealer around all the seams
If too much work for you throw a rug down lol

1/4" plywood seems thin for the floor. If it is not structurally a problem then just fill it to level it off and put a carpet over it.

They do sale a linoleum repair kit just do a google search I repaired mine at home after I dropped a hammer claw first. It turned out well

Do you want it to look like it was never damaged? If you aren't too concerned about the cosmetic result of the repair, epoxy will work just fine. If you want it to look like new, you can epoxy the plywood to fill in the dent, then try to heat the linoleum to soften it, and hot glue it back in place. If that doesn't look acceptable, you can try to obtain some replacement linoleum for a patch. Some times, you can find a small piece of linoleum you can cut for a patch under an appliance in an area that you won't see. You can also replace all the linoleum to make it seamless.

A picture is worth 1000 words of description, maybe. Any chance you can post one up of the damage in question?