If a cohen cohabits, lives with publicly and may even have had a child with a jewish woman, born jewish, can they marry afterwards?
What if there is a time in which they are not living together anymore?
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(The prohibited marriage for a Cohen that results from a previous prohibited intercourse of the woman making her a so called Zona. There are two other Isur, Chalala, not relevant for the question, and Grusha, divorced, see below).
Rambam Isure Bia 17.2
There are three women who are forbidden to all priests [by Scriptural Law]: a divorcee, a zonah, and a challalah. There are four [forbidden to] a High Priest. These three and a widow.
Bound by [the prohibitions applying to a High Priest] are one anointed with the oil of anointment or one who assumed his position by wearing the additional garments, one serving in that capacity, a High Priest who was appointed and then removed from the office, and a priest anointed to lead a war. All of these are commanded [to marry] a virgin and are forbidden to marry a widow.
Regarding unmaried woman, the Gemara concluded in yevamot 61b
The baraita cited above mentioned that Rabbi Elazar says: In the case of an unmarried man who had intercourse with an unmarried woman not for the purpose of marriage, he has caused her to become a zona. Rav Amram said: The halakha is not in accordance with the opinion of Rabbi Elazar.
It's ruled as the Halacha Rambam Isure Bia 18:2
Whenever a person has relations with an unmarried woman, even if she is a harlot who wantonly makes herself available to everyone, although she is liable for lashes, she is not deemed as a zonah, nor is she disqualified from [marrying] into the priesthood. For she is not forbidden to marry [the people with whom she engaged in relations].
[When, by contrast, a woman] engages in relations with a man with whom relations are forbidden by a negative commandment that is universally applicable - the transgression is not specific to priests - or with whom they are forbidden by a positive commandment, she is forbidden to marry him, she is a zonah. Needless to say, [this applies if she engages in relations with a man] who is forbidden to her as an ervah, a gentile, or a servant.
For a great Priest it's different but there is no great Priest nowadays.
If they leave for a time it's not a divorce that makes her Grusha (divorced, an Isur Kehuna) because there is no divorce if there is no marriage, even miderabanan (for Yevama a get has some rabbinical issues but it's not linked to our subject).