my child tripped on a badly covered hole in the school playground and broke her wrist, who is responsible and do we have a case for compensation?

You probably do have a case for negligent supervision and premises liability/failure to maintain and repair, but you need to act fast. There is a 90 day limit that runs from the day of the accident to serve the school district with a notice of claim. If you don't do that, no matter how good the case is, it will be barred by the statute of limitations.