It is called child only welfare, what ever they call it in your state.
The reason we don't do this is:
If the amount they collect from him in any given month is more than they pay you, they keep the rest so they can average it out for the rest of the time.
If he pays you upon getting a job they still only pay you the welfare amount. In Oklahoma it $88 per month, doesn't matter how many kids you have, it's only $80 something dollars.
We get $240 on one child regularly, the other child gets nothing from his dad of the court ordered $170 per month, and my daughter pays nothing of her court ordered $170 per month per child, $340 per month total.
If we applied for the child only TANF, Temporary Aid to Needy Families, then they would take ALL of the child support paid to us and distribute the $88 per month, keeping the other hundreds. It is not a fair system.
To me the better deal is the child support enforcement taking the parents income tax refunds and disbursing those funds to us after the IRS gets theirs. Plus the child support the one dad pays is enough to buy our groceries for the month and a little extra. I work 3 part time jobs to pay for the kids classes at the gym/dance studio, the YMCA, soccer, BMX. etc....