Margaret,
Death registration only started in Ireland in 1864, so if Francis or Ann died before that there’s probably no record. I searched the records 1864 onwards. This was the only Francis death in the Dungannon area that I could fine. A widower who had been a labourer and died in Dungannon workhouse:
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_returns/deaths_1880/06461/4858145.pdf
If the admission records still exist you might find out a little more about him from them.
Here’s an Ann McMahon death in 1878. She was a farmer’s wife and they lived in Altaglushan. I haven’t been able to work out who her husband was, though it probably wasn’t a Francis, as I can't find a Francis death to match it.
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_returns/deaths_1878/020515/7199059.pdf
Dungannon is in the RC parish of Drumgath. Their baptism records start in 1821. I have searched them on Ancestry, but don’t see any mention of Michael or of other children born to Francis & Ann. Nor did I see there marriage. The records are also viewable free on the nli site, in case you want to check them yourself.
http://www.nli.ie
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. You might want to try them or, if you have already tested, you can transfer your results to them for no fee.
The North of Ireland Family History Society is running an Ulster DNA project in conjunction with FTDNA and can offer testing kits at a reduced price. http://www.nifhs.org (Go to DNA project on the website).
