Hi on this linkhttp://churchrecords.irishgenealogy.ie/churchrecords/search....
There are 6 children
Catherine born 1859
George born 1862
Mary born 1865
Nicholas born 1858
Sarah born 1867
Catherine born 1868
This is the main church you should contact
Ss. Peter & Paul's Church http://www.corkandross.org/parishes.jsp?parishID=59
Also in 1851 there was a William Kearns living in Paul Street Corkhttp://www.failteromhat.com/griffiths/cork/stpaul.htm
Also I found this a marriage in 1767 from same parish
1767, 28 November: William Kearns, master builder and architect of Cork, son of William Kearns, married Jane Plaince. (Cork Chronicle, 30 Nov 1767) (IAA) http://www.corkrecords.com/StPauls/ParishRegisters.htm
Not sure if the last two William Kearns are any connection but all within same close area of the City to your family.