Note: I asked this question ~two weeks ago with no response. I realize answering it may need a historian with a fairly narrow band of expertise, so here goes attempt #2:
For context, I'm reading Immigrants Against The State by Kenyon Zimmer, about Italian and Jewish Anarchists in America. There seems to have been a comparatively high proportion of leftists in those two groups, as well as German immigrants, compared to other large immigrant populations to the US. Thinking about it, I'm surprised that there was not such a radical strain in the Irish immigrant community (and later generations of Irish Americans) - especially considering the fairly strong socialist movement in Ireland itself.
So - am I off base? Was there a strong streak of left-wing radicalism among Irish immigrants?
It all depends on the area. There were Irish Miners in the U.S. through Appalachian Mountains. The Molly Maguires were a sort of Irish gang/union that fought it out with mining companies in Pennsylvania in the 19th century and we're also active in Ireland.
There was a stereotype that Irish workers were backwards and reactionary because of Catholocism, but this was promoted by politicians to keep workers divided, primarily.
Noted Irish Socialist James Connolly was a fellow traveler of the U.S.'s Industrial Workers of the World. Noted IWW organizer and later Trotskyist leader of the American SWP, James P. Cannon was the son of Irish immigrants as well.
One thing that fascinates me about this period is how brutal ethnic competition and violence was and yet how committed and successful if fleetingly many socialists were with regards to the fight for integration of the labor movement in the U.S.