There is a Patrick J Dorgan living in Manhatten in 1910 Census
http://search.ancestry.co.uk/cgi-bin/sse.dll?MS_AdvCB=1&...

And a WW1 Record too for a Patrick J Dorgan
http://search.ancestry.co.uk/cgi-bin/sse.dll?db=WW1draft&...

There is only 5 Patrick Dorgans born in Cork between 1880 and 1888, https://www.familysearch.org/search/records/index#count=20&a... maybe if you got the cert for the 1882 and the 1887 ones it might give you more of an idea when searching the American records