Is a wife obligated to stay with her husband who engaged in an illicit relationship that resulted in pregnancy, and is she obligated to raise the child if custody is awarded to the husband?

The husband must immediately repent from adultery. If he does not repent, the wife has the right to seek divorce or khul‘, especially since his continued adultery has negative effects on the family. As for the child born of adultery, the wife is not legally obligated to serve or care for him unless she agrees to do so. If the husband repents, it is preferable for the wife to remain in the marriage and help in caring for the child.