Hi Fran,
If you have access to ancestry there is the Woodward-Shepley public member tree has some the details you gave but has the wife as Catherine Kelly.
rootsireland.ie has the marriage record of Patrick Gibbons and Catherine ? in 1826 in Mayo. When I used Burke or Kelly in the search neither worked. rootsireland is a pay site so you would need to purchase the record in order to get Catherine's last name and the parish. It would also give you Patrick's and Catherine's parents' names so you could go back another generation.
rootsireland.ie also has a record of the church baptism of Patrick Gibbons in Mayo in 1804.If you purchased that record it would also give you his mother's name and the parish the baptism took place in.
rootsireland.ie does not have a baptism/birth record for either Catherine Kelly or Burke born in Mayo in 1804 + - 5 years.
Col