May a Kohen marry a girl who has a gentile father but Jewish mother?
A girl who was born from a gentile father and Jewish mother is considered Pagum/blemished for the Kehuna. This applies whether the mother was married or single at the time she conceived from the gentile. It is thus forbidden for a Kohen to marry such a girl.
Bedieved and time of need: If a Kohen transgressed and married such a girl, he is not required to get divorced. [In certain circumstances however, and under the jurisdiction of a Beis Din, it is possible to allow a Kohen to marry such a woman. Thus practically, in all cases of doubt one is to turn to a Beis Din to review the case and abide by their decision.]
 Michaber E.H. 4/5 and 19 and 7/17; Mishnah Kiddushin 66b; Rishonim brought in Rif Yevamos 45b; Rosh Yevamos ibid 36; Chelkas Mechokeik 4/3; Beis Shmuel 4/2
Other opinions: Some Poskim rule it is even initially permitted for a Kohen to marry a woman with a gentile father [Nemukei Yosef on opinion of Rif Yevamos ibid; Teshuvos Rama 69 and Binyamin Zev on opinion of Rif; Implication of Rambam Issurei Biyah 15/3; brought in Beis Yosef, Beis Shmuel and Chelkas Mechokeik ibid and Birkeiy Yosef 4/14, however Mishneh Lemelech 13/8 learns the prohibition applies even according to the Rambam]
 The reason: This is learned from an Almana, as if an Almana’s child with a Kohen Gadol is invalid for Kehuna then certainly a gentiles child with a Jewess is invalid for Kehuna. [Rif ibid]
Biblical or Rabbinical: Some Poskim rule the above prohibition is only Rabbinical. [Rameh Mipuno 124; Rav Akiva Eiger 91; Rashal Yevamos ibid 38]
Conversion: Some Poskim rule that the daughter of a gentile father and Jewess requires conversion. [Yearos Dvash Derush 17, brought in Pischeiy Teshuvah E.H. 4/1; Avnei Nezer 16 on Tosafus Yevamos 16b] Practically we do not rule this way. [Avnei Nezer 16; Yeshuos Yaakov 4/8; Beis Yitzchak 26; Achiezer 21; Chazon Ish 6/7]
 Michaber ibid
 Chelkas Mechokeik 7/26; Beis Shmuel 7/39; Ramban Yevamos ibid
Other opinions: Some Poskim rule that the girl is a Safek Chalal, and hence it is forbidden to remain married and they must divorce. [Rosh ibid; Beir Hagoleh on 4/5; Implication of Michaber ibid, as brought in Birkeiy Yosef 4/19; See Rashal Yevamos ibid 38]
 The reason: As such a girl is not deemed a Chalal, but merely a Pagum. Likewise, there are Poskim that are completely lenient even initially. [ibid]
 Such leniencies have been awarded by certain Rabbanim and Batei Dinim in a case that the Kohen does not have strong testimony of his Kehuna, or if the couple are already living with each other for many years, or have already been married in a secular court. Other Rabbanim and Batei Dinim however contest against these leniencies.See a case of allowance brought in Yabia Omer 9/7; Shema Shlomo 5/8 and Habracha 4/25 by Rav Amar; Rav Mashash in Shemesh Umagen 3/54; See cases of prohibition in Igros Moshe E.H. 1/5; Hiafa Beis Din http://www.daat.ac.il/daat/psk/psk.asp?id=624;