The 1st image for James is from the links that Dundee posted in reply#7 and again in reply#11.
Mary's death image is too small.
Familysearch links are fine anyone can open and quicker to copy and paste than attaching images.
Earlier Shanreagh asked for a link to the 1860 census. The 1870 & 1880 being in Dundee's reply#7. Eldest son is the most important to know as likely his father's name.
In your initial post you said things like "I have located a few documents"
"Married to Mary Catherine Baumy Price on 7/20/1854. I have a copy of the marriage license. She was a widow that was married to a James Barrow on 7/2/1852".
If helpful you can still search on Ancestry and eg Findmypast with logons even if can't view images, Ancestry tends to be more restrictive in what is in their textural summary than some other sites.
I find Familysearch rather clunky and more time consuming to search with or consuct a tree on, it is really internally within the USA that a sub might ease research as World won't help in Ireland pre 1860. It is however the info relating to the people born in Ireland that are of most use, not the younger children and events in the 1920-30's.
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1870 & 1880 census:
James Haskins 51 (1819) born Ireland 1880 James deceased
Mary Haskins 37 (1833) born Ireland 40 as Hawkins
Thomas Haskins 17 (1853) born Louisiana
John Haskins 14 24
---9 year gap---
Mary Haskins 5 15
William Haskins 3 13
Catherine Haskins 9 mo [likely deceased]
There are 2 new children that I didn't know about. It looks like they had 5 children
1870
https://www.ancestry.com/search/collections/7163/records/326648061880
https://www.ancestry.com/search/collections/6742/records/401778041900
https://www.ancestry.com/search/collections/7602/records/19775601 Mary Hawkins (61, TEN children, 3 living) at 2305 Louisana Avenue with son-in-law James & Mary Johnston - explaining why only the 3 mentioned on her death notice. Mary 68 in the 1907 death index.
James married to Mary Catherine Baumy formerly Barrow, nee Price on 20 Jul 1854 in USA
based on 1870 census that is - AFTER Thomas's birth.
Mary was a widow at marriage, is Thomas her son or was James a widower also?
A Familysearch or Ancestry tree link might be the quickest so everyone can see the full picture.