Assalamualaikum warahmathullahi wabrakathuhu,Nearly 1 year Before my dad passed away he got promise from my mom not to get married my sister to one of my cousin brother, but they had an affair for more than 6 years, and it's in full deep in between them so is it possible to get married them both and give kaffara for the promise what my mom did?

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

Your mother has no right to prevent her daughter from marrying someone whom she agrees to, as she has no guardianship over her. As for her having sworn not to allow her to marry, you should be kind to her and guide her in a way that suits her position as a mother and tell her that there is no Sharee'ah impediment to recanting her oath as breaking an oath is permissible, in fact, it is preferable if there is an interest in it, such as in her case. If she responds positively to your direction and breaks her oath, then let your sister get married and offer an expiation for your mother’s oath. But, if she does not respond, there is no objection to allowing your sister to get married, in which case, your mother would break her oath and must expiate for her oath. Still, care must be taken to please your mother and ask her for forgiveness, and to explain to her the daughter’s need to marry whom she loves. For more benefit, please refer to Fatwa ">49159.

Allah knows best.