I and my cousin like each other and want to get marry, we are not in any kind of haram relationship. Because of some family issues his mother doesn't wants him to marry me and has fixed his marriage with some other girl, but I still pray to get married to him. My question is, is my dua to marry him illegitimate? Can I still pray for marrying him?
If yes can you please recommend me any dua or amaal so that I get in nikah with him.

Yes you can pray to marry him, but it will only happen if it is what is best for both of you.