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.
If the matter is true that your father has a relationship with that woman, then he is sinful whether or not he committed adultery with her. The fact that the people know about this relationship does not make it permissible for him to maintain it.
Therefore, they must be advised to repent to Allaah. Then, if he wishes to marry her, and she is a Muslim or a woman from the people of the Book, he should do so according to the Islamic conditions, like the presence of a guardian and the presence of two witnesses. However, he should immediately separate from her; for more benefit please refer to Fatwa 94791.
You should keep ties with your father and be kind and dutiful to him; your father being dissolute is not a sound excuse to cut relations with him. However, your husband is not obliged to keep ties with him, and if there is any benefit in reforming him by deserting him, he may do so.
As regards that woman, she should be reminded about the evil she is doing and she should be prevented from visiting you. Nonetheless, if you visit your father, do not be kind to her, as this may deter her.