Does the previous support order address these types of costs? That would control. You both can argue the issue and a judge can decide whether it's appropriate to bring in the amended proceeding and how costs should be assigned. There is a possibility the judge is going to encourage the both of you to work it out through ADR if the previous support order is silent on the issue.
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Your sons criminal costs really hve little or nothing to do with child support hearings. Contact my office for free consultation 727-446-7659.