The Torah forbids a Kohen from marrying a divorcee, a “chalalah” – a woman born from the union between a Kohen and a woman forbidden to a Kohen – and a “zonah,” a woman who engaged in a forbidden relations. A girl born from a Jewish woman and non-Jewish man is not a “chalalah.” In the case under discussion, the mother had engaged in a forbidden relationship, and may therefore fall under the category of a “zonah,” but the daughter has not done anything to fall under this category.
There is a debate among the Rishonim concerning this matter. The Rambam (a”h) states that it is permissible for the daughter of a Jewish woman and non-Jewish man to marry a Kohen. The Rosh (a”h) and other Rishonim accept this prohibition as normative halachah. The Rif (a”h) and the Ramban (a”h), held that the woman in this case has the status of “Safek Pegumah,” – it is uncertain whether or not she is halachically eligible to marry a Kohen.
The Shulchan Aruch follows the view of the Rosh, that the daughter of a Jewish woman and a non-Jewish man may not marry a Kohen. However, Rabanim write that if such a woman did marry a Kohen, they may remain married, and do not need to divorce. Of course, a competent Rabbi must be consulted for guidance concerning such a situation. And in general, it behooves all people to determine the lineage of a prospective spouse before marriage. This is especially true when dealing with Kohanim, regarding whom the laws of marriage are particularly complex.