Is it obligatory to divorce a wife who allowed a man into the house without her husband's knowledge and has repeatedly committed mistakes, or should he be patient with her to avoid the collapse of their children's lives, and would the husband incur sin if he divorced her?

We do not know if your doubt is justified or not. If it is justified and there is proof of suspicious behavior on her part, then you must confront her and demand repentance from her. If she repents, then do not divorce her, as divorce is reprehensible without a valid reason. However, if she persists and you are certain that your suspicions are well-founded, then you may divorce her, and it is recommended to do so.