How can I repair a puncture in a hollow-core wooden door?
You can't.  Wood doesn't "pop back out".  The wood is actually broken.

If you just want to repair it structurally, then filling in with something like a suitable epoxy and sanding would work.

Getting it to look like the undamaged original will be impossible or very difficult at best.  This is due to the grain, and no replacement piece of wood having the same matching grain pattern.

The best try would be to very carefully get the pieces of the wood back into their original positions.  Then open the cracked areas again a little, carefully insert some wood glue, back to original position, and somehow clamp or hold in that position until the glue dries.  Then you still have to sand the result to make it smooth, and hope the few inevitable interruptions in the grain won't be too noticeable.