We know that cohen can marry the daughter of a convert and a born Jew. What is the mother is a convert and the father is a non-Jew? For example, the woman did not marry but conceived artificially using a non-Jewish donor or the woman converted but then conceived through relations with a non-Jew?

Shalom A Cohen cannot marry the child of a non Jew. Therefore if the convert had relations with a gentile the daughter cannot marry a Cohen. If there was no relations and the child was conceived through artificial insemination there are opinions that the child can marry a Cohen. Kol tuv