Here is what you are up against. Four men in 1902, 7 in 1900, and only one overlap. At least 6 possible Mary Leonards in Jersey City in 1900. A closer look at the 1905 census by you, me or someone else might turn them up, but I didn't find them at first glance. Ancestry.com had the index of the 1902 baptism, but no other suggested records for the parents.
1902 Jersey City City Directory
Burns,
Patrick, foreman, 180 16th
Patrick, laborer, 551 Grand
Patrick, laborer, 177 13th
Patrick J, bartender 147 St Pauls's Avenue
1903 Jersey City City Directory
Burns,
Patrick, foreman, 180 16th
Patrick, laborer, 177 13th
Patrick J, bartender 147 St Pauls's Avenue
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1900 Census, Jersey City
339 Johnson Avenue
Burns, Patrick, 38, single, laborer born in Ireland, arrived 1898, alien, boarder,
https://www.familysearch.org/ark:/61903/1:1:M9JT-LTG12 Spruce Street
Burns, Patrick, 23, single, brother of the head of house James J, arrived 1899, laborer
https://www.familysearch.org/ark:/61903/1:1:M9J1-YFL83 Clinton Avenue
Burns, Patrick, 30, single, brother to head Michael, born in Ireland, arrived 1887, foreman, steel steel works
https://www.familysearch.org/ark:/61903/1:1:M9JB-329246 Wayne Street
Burns, Patrick K, 26, single, son of head Bridget, a widow, born in New Jersey, a produce clerk
https://www.familysearch.org/ark:/61903/1:1:M9JT-9SX474 Henderson Street
Burns, Patrick, head, 34, widowed, born in Ireland, arrived 1880, freight handler
https://www.familysearch.org/ark:/61903/1:1:M9J5-NZS180 16th Street
Byrnes, Patrick, 25, single, son of head Henry, born in New Jersey, a foreman
https://www.familysearch.org/ark:/61903/1:1:M9J5-CJZ199 Warren Street
Burns, Pat, 24, single, brother in law to haed Pat McGovern, born in New Jersey, a barrel cooper
https://www.familysearch.org/ark:/61903/1:1:M9JP-64Dat familysearch.org, I found 6 Mary Leonards in the 1900 census in Jersey City, who were of childbearing age and single.
I don't see the child Patrick, age 3 or so, on the 1905 Census of New Jersey. I might look more closely later.