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.
It is not permissible for a Muslim man to have an affectionate relationship with a non-Mahram woman even if this relationship did not lead to Zina (fornication or adultery). The prohibition of establishing such a relationship with a married woman is even more confirmed as it represents a crime against her husband by violating his honor.
Besides, a man urging a woman to seek divorce from her husband so that he himself can marry her is committing a great sin, as this is spoiling her marital relationship with her husband. Abu Hurayrah narrated that the Prophet, sallallaahu ‘alayhi wa sallam, said: "He is not one of us who spoils the marital relationship between a wife and her husband." [Ahmad and Abu Daawood] Therefore, this is a grave sin.
Moreover, some jurists like those of the Maaliki School of jurisprudence, went so far in this regard as to adopt the view that the marriage of a man with the woman whom he spoiled her marital relationship with her husband is void. So, the matter is very serious. For more benefit, please refer to Fataawa 142925 and 92056.
Hence, if your father is as you mentioned in the question, then it is an obligation upon you to advise him to fear Allaah, to stop committing these evil matters and to completely cut off his relationship with this woman and with any other non-Mahram women.
He should also be advised to fear Allaah in regard to his wife and fulfill her rights which he is obliged to fulfill. In case he does not repent and persists in this condition, then it becomes permissible for his wife to ask him for divorce because the dissoluteness of the husband is one of the sound reasons for asking for divorce. Also, if a wife is harmed by staying with her husband, she has the right to ask for divorce as stated by the jurists . In case he refuses to accept her request for divorce, she is permitted to take her case to an Islamic court.
On the other hand, it is not a sin to hate your father because of his dissoluteness and this, in itself, is not considered as severing ties of kinship with him. However, you must not mistreat him by any manner of mistreatment, be it by words or actions, as in this case, you would be considered as cutting ties of kinship with him.
Furthermore, hating that woman because of her disobedience to Allaah is one of the qualities of the believers. Ibn ‘Abbaas narrated that the Prophet, sallallaahu ‘alayhi wa sallam, said: “The strongest bonds of faith are loyalty for the sake of Allaah and enmity for the sake of Allaah, and love for the sake of Allaah and hate for the sake of Allaah.” [At-Tabaraani]
In case a wife asks for a divorce from her husband without a valid reason, then she is sinful as this is prohibited by the Prophet, sallallaahu ‘alayhi wa sallam, as clarified in Fatwa 84309.
Nevertheless, a woman is not considered a prostitute just because she accepted money from a non-Mahram man. Prostitution is Zina (fornication or adultery); so in case he gave her this money because of this relationship (prostitution), then this money is not lawful for her.
Finally, we advise you to supplicate Allaah earnestly and as much as possible to bless your father with guidance and righteousness and make him steadfast on the straight path.
Allaah Knows best.