The issue will have to be rectified before the person can continue with the citizenship process. The possibility of deportation would not be likely nor would being incarcerated unless there is a court order in place mandating the payment of child support. If there is such an order the person will be considered in contempt of court and could be taken into custody and/or other legal remedies such as wage garnishment or seizure of bank account(s). It is possible the judge would give him or her time pay the arrearages and place the person on probation with the stipulation that any failure to comply to the order will result in being jailed.