What shall I do when I discovery my wife has relationship with another man?

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.

If it becomes evident to you that your wife has a relationship with another man, then it becomes an obligation on you to advise her and remind her of Allaah; it might be that she will repent and regret what she was involved in. If she repents, and it becomes clear to you that she has sincerely repented, then it is more appropriate to keep her as a wife and conceal her sin rather than to divorce her especially if you have children with her.

However, if it becomes evident that this relationship led to adultery, then according to the opinion of the majority of scholars you are obliged to make sure that she is not pregnant from this relationship by waiting one menstruation or until she delivers the child if she is pregnant. Nonetheless, the child will be traced to you when she delivers the baby as you are the husband. It is confirmed that the Prophet said: "The child is traced back to the husband who is legally married to the woman." [Al-Bukhari and Muslim] You have to protect her, conceal her sin and do not allow her to be in seclusion with marriageable men or to mix with them, or do anything that might lead her to such an illegal relationship. This is of course if she repents. However, if she does not repent, then there is no good in keeping her as a wife. Rather, it is not permissible for you to keep such a wife if she does not repent whether she committed adultery with him or not, because it is likely she will cheat you and bear children who are not from you.

Allaah knows best.