If the back child support debt is still listed as such by the government officials there is a chance your tax refunds will be offset to pay them.
The guidelines for round two of the stimulus payments states "no offsets". However, the IRS has not published guidelines on how they will handle government offsets from tax returns with stimulus credits.
Here is a link with more information on the stimulus program. See the section on "Receiving my payments" for information on offsets.
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The second round payment is exempt from offsets, per the law that created the payment. However, that will only apply to $1200 of your refund ($600 for you and $600 for your dependent). I don't think we know what the IRS will be doing about the rest of your tax refund. However, child support has been the first offset to apply and the last offset to be paused.
If you owe back support for a time when the child was living with the other parent, the state will probably take the point of view that you still owe that money. If you need to modify the child support order, you need to talk to a judge. If your state child services department considers you past due, it will be up to them to request an offset from the federal government, or cancel the offset.