Is there a specific supplication and time for invoking Allah for help in this adversity? And is there a sin incurred by preventing a mother from performing Umrah with her sister as a lesson, knowing that this is the mother's first Umrah and that it is done in a state of anger?

It is not permissible to establish an unlawful relationship with a foreign man, and one must repent from it and sever it. The questioner should denounce her sister for this and inform her of the Sharia ruling, and inwardly resent it as the lowest degree of changing the reprehensible. There is no specific supplication for this situation, but general supplications for relief from distress can be made. As for the sister's invitation to her parents for 'Umrah, it is a righteous act that should be encouraged, and there is no sin upon the questioner for preventing her mother from accepting if it is to deter the sister from her unlawful practices. The questioner should inform her mother of the Sharia ruling for her mother and the impermissibility of continuing it, and urge her to accept the 'Umrah and supplicate for her two daughters for righteousness and uprightness.