What should a wife do upon learning of her husband's repeated infidelity and illicit affairs, especially when there is a child between them?

Accusations of immorality or spying are not permissible except with clear evidence, or to repel an evil when suspicion arises. If the husband's adultery is proven, which is one of the major sins, the advice is to conceal it and help him to repent and become upright, and to urge him to pray and associate with righteous people. If he repents, then all praise is due to Allah. Otherwise, it is preferable to seek a divorce or khul' from him.