Can a mother leave the state even though the father is paying child support

Support has nothing to do with it. It's about custody and visitation. If there is a Court Order it has to be obeyed (and that includes allowing the father visitation). Only another Court Order can change the first Order.

If there is no Court Order the mother is free to go. The father can attempt to get an Order to bring her and the child back. If the move is seen to be an effort to deprive the father from having visitation the mother will have a problem.

It really depends on what the custody and visitation schedule says. Support and visitation are two separate issues. If the father has visitation or joint custody, moving will violate this and be subject to legal recourse up to and including loss of custody by the mother.

If there is no visitation or custody order, the mother is free to do as she chooses.

We really need more information to accurately answer this question.