Thanks for the added information, it sounds like Patrick was old enough to return to Ireland on his own when his father died and his mother remarried, unless the story was embellished a bit. And when you find out the year of his enlistment, and where, that will be a big help.
Did Patrick Edward Burns have any siblings? Where were they born? Do you know where his parents married?
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My guess is that the sponsors at his baptism were Mary Leonard's brother and his wife, since she wouldn't have a sister with the same name as herself.
This is grasping at straws and doesn't provide much information anyway, but in case both Patrick (Sr) and Patrick Leonard (sponsor at the baptism) were boilermakers. Besides, this man's wife's name was Julia but there is a "daughter" in the house named Mary. The numbers for her, Patrick, and Julia's children don't really "add up"
1899, 1900, 1901, 1902, 1906 Jersey City City Directories
Patrick Leonard, boilermaker, 48 Bright Street
he's not in Jersey City in 1908 unless he moved
and changed occupations
also for him
1900 Census, Jersey City
48 Bright Street
Leonard
Patrick, 37, boilermaker, born Ireland, arrived 1881, naturalized, married 15 years
Julia, 39, wife, Ireland, arrived 1880, married 15 years, 12 children, 3 alive
Katey 14, daughter, New Jersey
Francis, 11, daughter, New Jersey
Mary, 22, daughter, New Jersey, married for 2 years, 2 children, 2 alive
https://www.familysearch.org/ark:/61903/1:1:M9JR-F3F-------
but no Patrick Burns, boilermaker, in 1900 or 1902 in Jersey City
1900 Jersey City City Directory
Patrick Burns
bartender 1 Hoboken Avenue
checker, 234 6th
peddler 218 2nd
Patrick F, bartender 246 Wayne
Patrick J, cigars, 594 Grand, house at 582 Grand
1901 Jersey City City Directory
Patrick Burns
laborer
639 Grove
129 Union
443 York
177 13th
cigars, 592 Grand, 593 Grand
of Brown & Burns, 747 Newark Avenue