What is the truth about "Priest" Vallon being the "father" of Amsterdam Vallon?

He could also have been a real Priest who had a secret mistress on the side. This was quite common in the past. Or maybe he got defrocked for offensive behavior, such as crime and womanizing.

I have a feeling, though, that it was just a nickname.

Desmond Tutu has a wife and 4 children.

The truth is that Liam Neesson is an actor. Think of it what you must.

Catholic Priests aren’t supposed to molest children either but plenty of them do. Maybe he just said to hell with it, maintained his clergy and had a kid anyway. Is that really so hard to believe?

Is it possible he sired a child before he became a priest?

I think it's something akin to #1 because he's not the person who delivers Holy Communion at the start of the film. The priest who delivers Holy Communion at the beginning is the same one we see later when St. Patrick's Cathedral is being built. I think "Priest" was a nickname Vallon acquired because he was a devout Catholic and dressed in black. In a similar sense, Monk wasn't actually a monk, but that was his nickname.

I got the impression that, like most relationships on social media, "it's complicated."

I think it's a combination, mostly of numbers 1 and 4.

My feeling is that he started out as a priest or was in-training to be a priest. He breaks those vows and has Amsterdam. This is the point where he leaves for America, disgraced by his actions.

Once in America, he does what needs to be done to survive, breaking with Christanity's principles more and more until he is the violent man we meet in Gangs of New York. While living in the Five Points, he is called "Priest" by those who know of his past. Perhaps the Monk started it? After a while, Vallon's social status brings him the leadership of the Dead Rabbits (or he founds the group) and he is called "Priest" now as much out of affection and recognition of his status of leadership as because of his past.

PS
I did a bit of digging just now, because this is one of my favourite films and you got my curiosity up.

I ran a search, asking the internet if Priest Vallon was based on a real person. As near as I can tell: no. Many characters in Gangs are real or composites, but Vallon seems more like he embodies attitude and provides needed character for Amsterdam and Bill.

I happen to have a thick "Making of Gangs of New York" book (including the shooting script!) with interviews, etc. I pulled it from my shelves and thumbed over the interviews with the screenwriters and with Liam Neeson. Neeson just says his character is an Irish immigrant - doesn't get too in-depth after that. The screenwriters and researcher said little about Vallon himself, outside of one writer. Jay Cocks says, "The first inkling I had of Amsterdam's character was his father's voice. I scribbled this thing down on a piece of paper: 'The blood always stays on the blade.' That line was the first thing that was ever written for Gangs of New York."

The shooting script opens with this line: "VALLON, a man in black, is shaving while his son, AMSTERDAM, watches in the shadows." Later, it says, "He fastens a white priest's collar around his throat."

Notable, also, is that, in the opening flurry of scenes that mark the introduction and melee of the film, only Bill the Butcher calls Vallon "Priest," although very few people (Amsterdam, Bill, and Monk) speak with him at all.

Later, after Bill stabs Vallon, the script says, "The Natives and Rabbits turn and look at the tableau of Bill holding the bloody priest." This is the only time the shooting script's stage directions refer to Vallon as "priest," but that's not really definitively saying he is one.

This site says he immigrated first and then fathered Amsterdam, but I don't know what its source is:
https://historica.fandom.com/wiki/Priest_Vallon