We are working on a kitchen remodel for a cabin that was built as a summer home, and we now live in it permanently. We already started tearing out the wood floors, and are finding they were glued to the plywood subfloor. The subfloor is just getting shredded trying to pull out the boards. Do we need to replace the entire subfloor in the kitchen area? Or can we lay a thin sheet of plywood over it and then lay new 3/4" wood floors?

Yes. Overlay the old, torn up subfloor with new 3/8 or 1/2" plywood.