It is possible (but not certain) that your child support is being calculated on the basis of the income left over after payment of previously-determined support obligations. There is no certain way of doing such a "multi-family" calculation, which I explained in an artilce that you can review here.
As to your suspicion that the support payments are artificial because the couple is living together, you are stuck with either old-fashioned detective work (as in a private investigator) or filing a motion and having lawyers get utility bills, etc., to figure it out.
As to welfare fraud, you can make whatever complaints you want to the public agency, which may or may not investigate -- but think through clearly what the effect would be on you and your child if the state did take action against the other parent. And I think the word you wanted was "prosecuted" (again, that may not be in your child's best interest); I doubt any combination of these facts will result in anyone being "executed."
Answered on Oct 08th, 2016 at 1:36 PM