the mother has signed papers with child
support enforcement years ago to drop all
back and forward child support
Probably can't even begin to answer your question without knowing the terms and conditions of what the mother signed and why it was signed in the first place.
can the mother go through the courts to get
the back child support she has signed to
Anybody can go "through the courts" for anything. Successfully doing so is another story. Courts are loath to overturn a written agreement where the parties or party had reason to make the agreement and knew what they were doing, but I suppose it could happen.