Your bicycle mechanic should be able to remove the old single speed freewheel and order a new one for you. They have a special tool which fits into the four notches.

However, I also notice that the hub seems to be missing a spacer. When you tighten up the wheel in the dropout, does the dropout press against the freewheel? That alone could be your issue. Your mechanic might be able to provide a spacer as well.