NY allows some sort of credit for that, on state taxes. Maybe some other states.
But there is no credit or deduction for child support on the federal return. Furthermore, just paying child support does not allow you to claim the child as a dependent.
The custodial parent has first priority on claiming the children on her taxes; regardless of the amount of support provided by the non-custodial parent. The IRS goes by physical custody, not legal custody. The non-custodial parent can only claim the child as a dependent if the custodial parent gives permission (on form 8332) or if it's spelled out in a pre 2009 divorce decree. Even if a divorce decree, dated after 2008, gives the non-custodial parent the right to claim the child, he must still get form 8332 from the custodial parent. A properly worded decree should require her to provide that form. https://www.irs.gov/pub/irs-pdf/f8332.pdf