I think someone has already mentioned looking for immigration data in the country of arrival.
Another tip is to look for your people on every census in their new country. This is important as some /one US censuses has a question asking 'how long have you lived in the US?' You can often use this to pinpoint when the family/person arrived in the US, where, who came with them and some times where they embarked. Where they embarked can also provide clues. Sometimes families from the north sailed, in early times from Londonderry, or later Liverpool.
Belfast
Cobh (in those days known as Queenstown).
Also when I'm starting to research I have a quick look at the 1901/1911 Irish censuses just to find out where the families of that name are mainly found in ireland. Sometimes you can find an elderly person with a first name of one of the children to the families in the US.
https://www.census.nationalarchives.ie/search/Looking at the Irish naming pattersn can be useful
A traditional naming pattern was often used by Irish parents until the later 19th century:
First son usually named for the father's father
Second son usually named for the mother's father
Third son usually named for the father
Fourth son usually named for the father's eldest brother
Fifth son usually named for the mother's eldest brother
First daughter usually named for the mother's mother
Second daughter usually named for the father's mother
Third daughter usually named for the mother
Fourth daughter usually named for the mother's eldest sister
Fifth daughter usually named for the father's eldest sister.
https://www.familysearch.org/en/wiki/Ireland_Naming_CustomsI have a US Census from 6/4/1860 that shows a James Hawkins, age 22 from Dublin Ireland living in New Orleans. Not sure if this is the correct James but his age of 22 would be feasible.
I have a migration form for the gulf ports region saying that a James Hawkins Jr arrived during 1846, age 7 and from Ireland. The age matches but not sure if this is the correct personCould you please provide a link to these please. Surely James Hawkins aged 7 in 1846 did not come by himself???
The 1870 census you have linked to has these children Thomas, John (14), Mary, William, Catherine (91/2). In view of the large gaps in ages between John and mary, william and catherine have you got a complete list of the children born to this couple - even if they subsequently died young as naming patterns could still have been used.
