What is the ruling regarding a mother's relationship with a man other than her husband, given that she was forced to marry the father and does not feel comfortable with him, and what is the solution to avoid scandal and negative impact on the children?

The Islamic principle is that a woman has the right to choose her husband. Forcing her to marry someone she does not desire spoils marital life. What the mother is doing—corrupting religious and worldly affairs, and causing scandal—is forbidden (haram). The Sharia has provided an exit for her by allowing her to take her case to court to annul the marriage, or to seek a divorce from her husband (khula'). You must confront your mother frankly and remind her of Allah and her husband's rights. If she responds, then praise be to Allah. If she persists, you must seek the help of her wise relatives to deter her. There is no excuse for this action of the mother, as disobedience is impermissible, and her action is not under duress. One must be content with Allah's decree, and the mother's wrongdoing must be denounced. If you undertake this, you will be rewarded, and the mother will also be rewarded if she repents. This is from the blessings of enjoining good and forbidding evil.