Hi Val
There's a gap in the Tyrone records 1814 to 1838 on www.rootsireland.ie/ so no luck with Margaret or the parents' marriage.
Here's another child though:
Name:Patrick HannonDate of Birth:
Date of Baptism:09-Mar-1808Address:
Parish/District:FEDAMOREGender:UnknownCountyCo. Limerick
Denomination:Roman Catholic
Father:Thomas HannonMother:Elizabeth BourkeOccupation:
Sponsor 1 /
Informant 1:Ml HartiganSponsor 2 /
Informant 2:Mgt Keogh
The R C registers are free online at registers.nli.ie/ & have Fedamore baptisms & marriages 1806-13 1814-22 &1826-33 &1833- 84 so you might want to try there if youm haven't already but it's hard going.
Here's the only record of Thomas Hannan around 1780:
Name:Thomas HannanDate of Birth: Date of Baptism:29-Jun-1783Address:ShanganahParish/District:KINGSTOWNGender:MaleCountyCo. DublinDenomination:Roman CatholicFather:Maurice HannanMother:MaryOccupation:Sponsor 1 /
Informant 1:William Martin
Sponsor 2 /
Informant 2:Mrs. Foley
Possibles for Elizabeth:
Name:Elizabeth BurkeDate of Birth:
Date of Baptism:19-Oct-1790Address:Not RecordedParish/District:CARRICK-ON-SUIR PARISHGender:FemaleCountyCo. Tipperary
Denomination:Roman Catholic
Father:Patrick BurkeMother:Maria CantwellOccupation:
Sponsor 1 /
Informant 1:Il IlSponsor 2 /
Informant 2:Anastasia Howley
Name:Elizabeth BourkeDate of Birth:
Date of Baptism:13-Oct-1796Address:Not RecordedParish/District:NEWPORTGender:FemaleCountyCo. Tipperary
Denomination:Roman Catholic
Father:John BourkeMother:Maria ButlerOccupation:
Sponsor 1 /
Informant 1:Ego DurochasSponsor 2 /
Informant 2:Eliz Lawlor
Col