A Non-Jewish Father
Although one whose mother is Jewish is unquestionably considered to be a Jew, one whose father is not Jewish is considered to be "pagum," "blemished". Such a person is subject to certain restrictions and disqualifications. According to some authorities, these restrictions are biblical in nature, while according to others they are rabbinical enactments. It is interesting to note, however, that there is an opinion that one whose father is not Jewish is also deemed to be a non-Jew regardless of the mother's status, though the halacha is not in accordance with this view.
The restriction that we will discuss in this chapter is the ban for a woman whose father is not Jewish to marry a Kohen. As the Shulchan Aruch writes:
If a Jew has relations with any of the prohibited women, the status of the child follows the status of the mother. And if any [of the prohibited men]… except for a mamzer… has relations with a Jewish woman, the child is kosher to marry into the community, except that the child is pagum to a kohen.
Most contemporary authorities concur with this view and rule that although the child of a Jewish mother and non-Jewish father is deemed a Jew, a female born from such a union is not permitted to marry a kohen.
On the other hand, there have been authorities in the past that permitted such marriages. There is also a view that although such a marriage is ultimately permitted, it is to be discouraged. According to most authorities, a Kohen who has gone ahead and married such a woman is not required to divorce her. It is not completely clear, however, whether or not the children of a Kohen who married a woman whose father is not Jewish are to be considered kohanim.
Most authorities rule that a rabbi should not officiate at the wedding of a Kohen and a woman whose father is not Jewish. Some authorities permit a rabbi to officiate at such a wedding if the couple was not aware of this halacha when they first became involved with one another. There are also a number of other considerations which have led some contemporary halachic authorities to rule leniently in such a situation, as well. Nevertheless, due to its severely problematic status, one should go to great lengths to avoid such a marriage.
 Yevamot 45a.
 In addition to the restriction of a woman with a non-Jewish father.
 Shu"t Rabbi Akiva Eiger 91; Beit Meir, EH 6:17; Igrot Moshe, EH 1:5.
 Tosfot, Kiddushin 75b; Kitzur Piskei Tosfot 142. See also Chemdat Shlomo, EH 2,3.
 Yeshuot Yaakov, EH 4:8,10; Avnei Nezer, EH 16; Achiezer 21; Chazon Ish, EH 50.
 For additional restrictions and disqualification for such people see: Yashiv Yitzchak 11:37.
 EH 4:5. See also EH 4:19, 7:17.
 Yevamot 45a; Rosh; Yevamot 4:30; Tur, EH 4; Gra, EH 7:54.
 Rambam, Hilchot Issurei Biah 15:3; Pitchei Teshuva, EH 4:1.
 Yevamot 77a; Yam Shel Shlomo 18; Shu"t Harama 18,61,69; Rivevot V'yovlot 2:409.
 Beit Shmuel, EH 4:2, 7:39; Chelkat Mechokek, EH 4:3, 7:26; Pitchei Teshuva, EH 4:3, Seridei Aish 1:71.
 Ramban, Yevamot 45a; Minchat Yitzchak 2:131.
 Igrot Moshe, EH 1:15.
 Shemesh U'magen, EH 3:58; Shema Shlomo, EH 5:8. Cited at: http://www.koltorah.org/ravj/A_Kohen_Marrying_the_Daughter_of_a_Jewish_Mother_and_a_Non-Jewish_Father__2.html
 Yabia Omer, EH 9:7, 10:14