First, if there isn't a support order, then there is no unpaid support. If there IS a support order it just needs to be enforced, a suit is probably not necessary. Second what is a DHHR program? Third, ANY question on law needs to include your general locale as laws vary by area.

Yes you can either ask the courts to enforce the current order ( only way you can have back support)

Or you can file for it ( and should have already)

... Second what is a DHHR program? ...

If it is some sort of welfare, child support may be assigned to the state, in which case O.P. couldn't sue herself.