It sounds like you are under a child support order and are not the parent with the right to designate the residence of the child. If this is correct, then you are obligated to pay child support each month regardless of the visitation ordered. This will not be refunded to you. If you feel the child support is too high in comparison with your income, you should consult a licensed attorney in your area to talk about the possibility of petitioning the court to modify the existing order.
Child support orders generally require one party to pay child support year round- regardless of the periods of possession. However, if the parties agree, you can modify the order to change the obligations- for example, if the other party will agree to give you credit for your month of possession- you can likely modify the agreement to reflect that. Or if you believe you are paying too much in support (maybe you have taken a reduction in pay) you can seek to modify the amount of the monthly payments to reflect your current income. Having a good family law attorney by your side will help advise you on the best course of action.