I searched Find My Past for any record for George or Mary McChesney.
Apart from the Griffith's entry there is little of note.
Roots Ireland (subscription site) does not show any records pertaining to George McChesney in Cavan.
What was the name of Mary's uncle who she travelled to Australia with. Do you know in what way he was her uncle? Is there a possibility Mr Mitchell was the brother of her mother?
What denomination were the family?
The 1911 census shows one result for George McChesney in Co. Cavan. Possibly related to your family.