When someone owes child support of $2500 or more, they are put on a list that denies them the ability to obtain a passport. Once adequate arrangements are made for payment of the child support, they can be taken off the list. However, it can take several weeks before that happens.
You should contact the State Child Support Enforcement Agency in the state where you owe child support and make sure that they contacted the U.S. Department of Health and Human Services (HHS) to inform them that acceptable arrangements for payment had been made.
HHS takes your name off the list of those who are in arrears of child support. They send notice to the Department of State Passport Services which enables you to apply for a passport.