Jim:
I checked the subscription site Roots Ireland and there are a number of John Burke baptismal records around 1810 in Co. Waterford and a few in that time frame for Co. Kildare. The problem is we don't have John's parents names so we can't confirm any of these records. Also, it is possible your John was baptized in a parish that no longer has records back to 1810.
Roots Ireland did not have any John Burke church marriage records to a Kenny in either Kildare or Waterford.
You may want to consider a general DNA test.
Roger McDonnell