This does not affect your taxes.  This seems to be a special provision for NY state taxes that does not change your claim on your Federal Return.  If he is current and qualifies (which you can see by clicking on this link:  https://www.tax.ny.gov/pdf/current_forms/it/it209i.pdf), he is entitled to claim this Credit for New York if he qualifies, but this does not prevent you from claiming your children.  

From a practical standpoint, this is also good for you:  if he is able to claim your children for this credit, he also has incentive to keep current on his child support payment, which is a plus for you and your children.