Hi Bruce . . .
Welcome to Ireland reaching Out . . . and Tipperary . . .
I began my search . . using " Roots " (paid) . . . for
Mathew Hayes . . b.1802 Tipperary . +/- 5 years . . . Nothing . . .
https://registers.nli.ie/parishes/0804
If you have the time and the patience . . the above site leads you to Nenagh R.C. Parish records . . . for those years . . .
again . . . I searched for the BIRTH . . of Ellen Berkey . . . b. ??? 1800 . . . Nenagh . . +/- 10+++
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Next I searched for a MARRIAGE . . of BOTH . . . 1830 . . . Tipperary . . . +/- 10 years . . and FOUND . . (see below)
( witnesses to this marriage were . . . John Ryan and Bryan Farrell ) . . Brides name spelling . . . is prob. a misspelling of Berkey ( this is a common Tipp. surname )
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DEATH . . of ELLEN . . is 1874 ..Nenagh . . . ( see below ) . . . note . . 1874 . . 67 y.o. . . . . BORN 1807
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Next I searched for the DEATH of Mathew Hayes . . . who died 1864 . . . Nenagh . . NO record . . .
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I went back to BIRTH . . of Ellen in 1807 Nenagh . . . NO record . . .
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MARY 1831 . . . (see below )
MICHAEL 1834 . . (see below )
MATHEW . . 1837 (below)
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There is a TAX roll for the period 1823 - 1837 . . . hundreds of Hayes . . . Tipperary
http://titheapplotmentbooks.nationalarchives.ie/search/tab/results.jsp?surname=Hayes&firstname=&county=Tipperary&townland=&parish=&search=Search&sort=&pageSize=&pager.offset=0
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NEXT . . . another TAX roll . . . GRIFFITHS . . . 1847 - 1864 . . HAYES ..Youughalarra . . . showing . . .
http://www.askaboutireland.ie/griffith-valuation/index.xml?action=doNameSearch&Submit.x=32&Submit.y=8&familyname=Hayes&firstname=First+Name&baronyname=&countyname=TIPPERARY%2C+NORTH+RIDING&unionname=&parishname=YOUGHALARRA
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BRUCE . . . I hope some of the above is useful . . . to your continued searching . . . Best I could do . . .
I should tell you there is a Professional Search available . . in NENAGH . . ( Paid svce. . . not cheap )
https://www.tipperarycoco.ie/heritage/genealogy-centre
Cheers . . . Eamonn.