New Custodial parent owe ex-custodial parent arrears

This is in New York State. Child lived with mother, who left him alone at his grandmother's house one day and now refuses to see him. Child moved into father's house that day. Father petitioned custody and child support together. Father given temporary physical custody at custody ruling, and mother only allowed to see child under supervised visitation because of problems with drugs, homelessness and suicidal tendencies.

Child support ruling coming up at the end of the month, and father's support deficit building up to over $2000. Some of this debt is arrears from when mother had custody before child came to live with father, because of incorrectly posted cash payments. Father had driver's license taken away, and receiving letters threatening him with jail. Child support garnished large chunk of money out of bank account and also a saving account intended for child's education. Father discovered that bank accounts were empty when trying to buy groceries to feed child. When father went to child support office, woman working told him that the deficit will be owed to mother no matter what, cannot be dismissed, cannot be credited toward mother's support obligations, even when (it's not a matter of if, it's when) she owes child support. Is that REALLY true?

The worker also said that if the court determines that mother shouldn't have the money, the deficit will still be owed and disbursed among other custodial parents who haven't received their due child support. Thought child support legally belonged to the child? How can this child's money be given to another unrelated child somewhere else in the state? Father makes $10,000 annually and needs that money for food and daycare, not supporting unrelated children who have nothing to do with him or a non-custodial mother who is not allowed to see the child by herself.