Is Hajj valid and does it permit the continuation of an illicit relationship (non-marital) with a non-Muslim man, knowing that this relationship is recognized in North American countries as if it were a marriage?

You must know that this cohabitation is explicit fornication (zina), which is one of the gravest sins. Severe warnings concerning it have been mentioned in the Prophetic traditions (Ahadith), such as the vision of fornicators in a hole like an oven in a dream, and their being suspended and disfigured in another vision. Allah has prescribed the punishment for the fornicator as one hundred lashes for the unmarried, and stoning for the married. Therefore, you must hasten to repent to Allah and sever ties with this man.

Secondly: It is not permissible for you to marry this man unless he declares his Islam and adheres to its rulings, because the marriage of a Muslim woman to a disbeliever is Haram (definition) by consensus, and is considered fornication. If you are living with him in this forbidden relationship, then know that you are facing two choices: either to prioritize Allah and the Hereafter, or to prioritize this sinful life that will not last.

Thirdly: Persistence in sin can lead to the hardening of the heart and prevention of repentance, and dying with a bad end. Therefore, you must return to Allah with repentance and remorse, and frequently supplicate to Him to accept your repentance and forgive your sin, for He, Glory be to Him, loves those who repent and rejoices at their return.