Is a Kohen allowed to marry a woman who is the daughter of a Jewish woman and non-Jewish man?
According to Shulchan Aruch, Even Ha'Ezer 7:17 and commentaries there, they should not marry. However, if they did so, they may remain together and need not divorce.
This issue is disputed both in the Gemara and in the Rishonim. The ruling of the commentators of the Shulchan Aruch is that lechatchila a kohen should not marry her, but if he did marry her he is not required to divorce her. There have been cases where authorities have been lenient that even if a kohen was only engaged to her, he may marry her.
For further discussion about whether a Beit Din may sign off/administer the wedding, see the Machloket between Rav Moshe Feinstein (Iggerot Moshe Even HaEzer vol 1 Siman 5), who forbade it, and Rav Shalom Meshash (Shemesh U’Magen vol. 3 Even HaEzer 58) and Rav Shlomo Amar (Shema Shlomo vol. 5 Even HaEzer 8), who permitted it, as discussed in by Rav J.D. Bleich, Rav Elyashiv Knohl, a Beit Din in Haifa, and Rav Howard Jachter. Rav Ovadia in later years was mattir outright based on a double Safek: whether he's really a kohen or not and if the Halacha follows the Rishonim and Shulchan Aruch who hold a Bat Nochri is pegumah or like those who disagree. (Yabia Omer vol. 11 Even HaEzer 34)