Your receipt of public benefits can only improve your position with the court as it is against public policy to allow a parent with the child to be a public charge while there is another parent failing to contribute support.
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You'll get the municipality to assist you to get him to pay. As the primary caretaker, you'll win custody. He is obviously getting zero legal advice so he is walking into an ambush.
The court in deciding factors for custody does not focus on any one particular factor. The court looks for the best interest of the child or Children. I would not be so bold via this forum to give my personal opinion on who should and will win custody, only the Judges opinion maters.
That beings said what we lawyers frequently say please avail your self of legal counsel, despite the possible cost to you. The lawyer can take your valuable impute and information and "package it " and reinforce it so the judge has the best information to make his or her decision.
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Being on food stamps will have no effect on your petition for upward modification of child support. However, being on Medicaid (I'm assuming you meant Medicaid and not Medicare since Medicare benefits are for people over the age of 65), will most likely have an effect since you will eventually be forced to assign your child support to the county you reside due to the county paying for your child's medical insurance. You need to have the court force the father to immediately place the child on his medical insurance so that you can take your daughter off Medicaid. This way you get to keep the food stamps and his child support. If you have your daughter on public assistance in addition to food stamps and Medicaid, you will also have your rights to child support assigned to the government.
It is possible that the county you reside in will not go after the father and that you will be able to have your daughter stay on Medicaid while still receiving child support. However, if they do file a petition against him, the child support will then be given to the county instead of you.
As far as custody is concerned, most courts believe that child support and custody are two separate things, however failing to pay child support, never looks good to a judge.
Advice on this forum is for informational purposes only and should never be mistaken as a substitution for legal advice. Answering a question does not create an attorney client relationship. If you need legal advice, you should consult or retain legal counsel.