Generally you should be able to accomplish this through your office of child support enforcement. After paternity is established he could seek visitation rights.
I am only licensed in Arkansas. No part of this answer should be construed as creating an attorney client relationship.
You certainly can seek child support through the state child support office. Since you gave us no financial information, there's no way to answer how much.
As for custody, anything is possible, but he'd have to go to court and file to legitimize the child, and given the likely consequences to his marriage, he may not. No one here is a mindreader to know what he might seek. If he seeks any rights, see a lawyer.
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I agree with my colleagues. You can get child support from him but, without more information, it is impossible to tell you how much. Given that he has other children, more work may be involved to determine what he pays for his other child and his expenses for the children with his Wife because, depending on the circumstances, he may get a credit for those payments which may decrease his child support obligation to you. If you only go for child support, he does not have legal rights to the child. However, it is possible that after you ask for child support, he will file to legitimate to get legal rights and ask for parenting time and potentially custody.