What should I do about my mother's scandalous relationship with my cousin, this not being the first time, and my father constantly traveling?

You must prevent your mother from committing zina (fornication/adultery) with all your might. This is a form of righteousness towards her and your father.

Means of preventing her include:

1. Reminding her of Allah, the Last Day, and the punishment for those who commit zina. 2. Reminding her of repentance (tawbah) and its virtues. 3. Reminding her that this act brings disgrace and shame in this world. 4. Monitoring her and preventing her from mixing with those you mentioned. 5. Preventing her from going out if her going out is for that purpose. 6. Threatening her with informing your father or her family (her father or brothers) if she does not stop.