Margaret,
The normal spelling of the parish is Drumragh (pronounced Drumraa) and the town of Omagh is in it. I see Patrick’s occupation as farm servant on the immigration record so that would suggest his family lived in the rural areas around Omagh rather than in the town itself. Being a weaver and a farm servant are not incompatible. Most farm labourers in Ulster did weaving when there was no farm work required. It brought in a bit of extra income. The majority of Irish linen was manufactured in the counties of Ulster. Originally on collapsible looms in labourers cottages but later replaced by more efficient water powered weaving factories.
Unfortunately the local RC parish records don’t start till 1846 so you are not likely to find any record of Patrick’s baptism or his or his parents marriage.
Griffiths Valuation (1860) lists just one Denis Barrett in Drumragh. He lived in the townland of Aghadulla on plot 22a and also had had a third share of about 10 acres of land (plot 12). Numerous other Barretts in that townland including 2 more Patricks. Numerous McCaffrey households in the townland too. There was a tendency to marry the girl next door in those days, so it’s possible Aghadulla is where your Patrick originated.
Plot 22a in Aghadulla today is on the modern Tattysallagh Rd, near the intersection with the Harpur Rd, a few miles west of Omagh. (Road names in rural Ireland were only introduced in the 1950s and prior to that your townland alone was sufficient to identify you and get a letter delivered etc). There’s a modern house on 22a now but the area is still all farmland.
https://www.askaboutireland.ie/griffith-valuation/index.xml?action=nameSearch
Here’s a death of a Denis Barrett from Aghadulla in Omagh workhouse in 1891. He was a 61 year old bachelor, so not Patrick’s father but a relation perhaps. He’ll likely have gone to the workhouse for free medical treatment in their hospital, rather than because he was destitute. Suspect Patrick’s father Denis died before 1864 when death registration started.
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_retu…
No shortage of Barretts & McCaffreys in Aghadulla in the 1901 census. Take your pick.
http://www.census.nationalarchives.ie/pages/1901/Tyrone/Clanabogan/Aghadulla/
Aghadulla is 559 acres. Population in 1901 was 125 people in 32 households.
Possibly DNA testing may be a way of matching with others who have additional information about where the family originate. Family Tree DNA reportedly has more people with Ulster roots than any other company. That obviously increases the chances of finding a match. If you have already tested your DNA with Ancestry, My Heritage or 23&Me you can upload your DNA results to Family Tree DNA for free and then join the North of Ireland Family History Society DNA Project. Simple instructions on how to do this can be found here: https://www.nifhs.org/dna/uploading-your-dna/