Welcome back!
Do you know if Mark had any siblings? There is a baptism on the NLI site for an Andrew Duignan born to a father Patrick and a mother Ellen Halliday. I couldn't find a marriage for the couple. Andrew was baptised in Milltown on 30 March 1861. Sponsors were John Melvin and Catherine Duignan. This was the only baptism I could fnd for children to this couple.
An Andrew is also mentioned HERE on the 1911 Census in the townland of Dalgin, Milltown, Co. Galway, he is unmarried. It may be the case though that this is not the correct family as in 1901 Andrew is living with a mother Bridget HERE who is a widowand the birth dates are a bit off.
There doesn't seem to be an Duignan graves that fit your criteria in Kilclooney graveyard or in Kilgeverin Graveyard the two most local. The family might be buried in either place but the grave may not be marked or perhaps the headstone is too hard to read and hasn't been transcribed. There is also a Patrick Duignan HERE on the Griffith's Valuation but I can't be sure if it's your Patrick
Let me know what you think
Best wishes
Clare Doyle
Genealogy Support