Hi I don’t have anything particular on John but McMurrough Kavanagh is quiet a unique name in Ireland. I think they all come from Borris. There is an excellent article about Arthur a highly successful man and a lot of detail on his records. https://www.theirishstory.com/2012/12/21/arthur-macmurrough-kavanagh-the-limbless-landlord/#.X1qP6CXhSEe
Turtle Bunberry a historian of Kilkenny and Carlow has done some research on Borris House and McMurrough Kavanaghs you could look at some of his sources too.
It’s not an answer but it will certainly give you some connections to his name. I will look to see if Borris House has a collection of records and how to access them.
Best of luck