Alexander, it seems most Burkes were in the Civil Parish of Clones.Do you have the correct spelling for the Townland?(It would help, yes.)
There was one Burke family with a gap in birthdates (not that you can hang your hat on it), so here's their family line:
Father: Thomas Burke b. c. 1792 Married: Elizabeth Young at Guardhill, Killeevan Civil Parish, Clones Union, Co Monaghan Children (found): _ William Burke b 1814 Boughill, Barony Dartee, Clones Union, Co Monaghan / d before 1900 / Buried: Drumsaul, _ , Clones Union, Co Monaghan / Married: Sarah Longmoore _Thomas Burke b c. 1820 _Robert Burke b c. 1822 [Here's the gap] _George Burke b c. 1827 _Daniel and Joseph Burke b c. 1828 _Mary ELizabeth Burke b c. 1829 / d before 1900
Then there's another line: Father: Alexander Burke / Mo: Rachel Lewis _ James Burke b c. 1838 _, Monaghan / m. 1864 Feb 11 Aghabog, _, Monaghan to Prudence Simons
There are also BURK's and given the challenge of spelling names correctly (seldom met with consistency), one is left with perusing alternative spellings.The BURK's are also numerous in Clones Civil Parish.