I always fill nail holes after every move out. I specify in lease that tenant is NOT to do this. They can ruin your walls in a hurry if they do it wrong.
In lease I allow small holes but charge for anything over 1/4", a few bucks for small and $5 for 1/2" holes - after all they are permanent damage to the wall. ( just had someone move out that left 32ea 1/2" holes!)
The plastic thing in your wall hole is an anchor, you can pull them out with needle nose pliers and then patch the hole.