The RC parish records for Glencolumbkille don’t start till 1879 so you are unlikely to find any record of Cornelius’s birth, if he came from Meenachavey.
Griffiths Valuation for 1857 lists a Peter McCunnigan farming in that townland. There were only 7 properties in the townland so there’s a strong chance this is a relation. (The Mc & O prefixes on Irish names are detachable and were used and omitted at will. So that leaves you with Cunnigan which is a pretty close match for Cunningham. Don’t expect consistent spelling of names in Ireland in the 1800s). Peter was farming plot 2 in common with 9 other tenants.
One Cunningham family still farming there in 1901:
And in 1911: