That would depend on what your custody order says. If you have joint legal custody, then that means that you have an equal say in school, religion and medical decisions for your child. If your ex has primary legal custody then she can decide this without your agreement.
She may be able to go back and have the child support increased to cover the increased expense now, but she can not have you pay an increase to cover past expenses.
Recently my x placed our child in a private school and now several months later she wants me to pay for it. Does she have legal grounds to have this added to my already amount I am paying for support? This was not my idea nor my wishes