A tree online says that Owen Robert Clarke was baptized 13 March 1907 in St Monica's Church, Manhattan. You could try them for baptisms or burial records. They might be willing to help.
https://www.stelmo79.org/welcomeFor the civil records, I think you'd have to send for all the Clarke births in Manhattan between their arrival and 1910. I don't see them indexed with parents' names on familysearch or ancestry.com
I looked for births of Clark from 1904 to 1910 here, because the output is easy to search:
https://italiangen.org/databases/And I looked only for the grandparents' names, Owen, Rose, Joseph, and Margaret, based on this tree at familysearch:
https://www.familysearch.org/tree/person/details/LV46-XBHThis is what I found:
8 Margaret Clark and 2 Margaret Clarke
4 Rose Clark but no Rose Clarke
for Owen, only your Owen Robert and a boy born afterward, so that would eliminate him.
2 Joseph Clark and 2 Joseph Clarke born in 1906 or afterward, after their first Joseph James died.
But they could have named the child anything, and there were a lot unnamed.
------
the familysearch tree says that their son Joseph James was buried in Queens, 6 May 1906. familysearch says he was buried in Calvary Cemetery:
https://www.familysearch.org/ark:/61903/1:1:2719-9SYYou'd think they would bury the other child with Joseph James, so you could try the cemetery as well.
https://www.findagrave.com/cemetery/64107/calvary-cemeteryEdit to add: If you try the cemetery, tell them the name and exact date of burial for Joseph James. That will give them a starting place, and the other child might be in the same plot anyway. I doubt they will look through everyone named Clark(e) buried from 1904 to 1910.
