what exactly does the term 'dismissed with prejudice' mean in a child support case?

A dismissal "with prejudice" essentially means that the court considered the application, the reasons presented and found that, based on the merits, the application should be dismissed.  A dismissal "with prejudice" means that the issues raised have already been decided.  The litigant whose claims were dismissed is barred from attempting to revive the same issues in a future application to the court.