According to the website Society for American Baseball Research, “Ann Leonard, her youngest son, Andrew, and his two sisters, Ann and Catharine, fled to America, sailing from Liverpool on the ship Milan, which arrived in New York on June 13, 1848. They settled in Newark, where they were joined by the family’s three oldest sons, John, Thomas, and Richard. No record can be found of what happened to their father.”
It’s not clear where the name Andrew came from as being the name of Andy’s father. If Andy said that was his father’s name, that’s pretty reliable; if it was on his death cert, it’s not so reliable. According to the previous blurb, Andy was the fourth son. In Irish naming patterns, the third son is named after the father. So it’s possible Andy’s father was not named Andrew.
Note too that some websites presenting info about Andy say that he was born in Antrim.