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Clifden is one of the few Galway West RC parishes that has records back into the first part of the 19th century with baptismal and marriage records starting in 1838. I searched on the subscription site Roots Ireland and found an 1849 baptismal record for a Gregory Kane (see below) and I'm wondering if he would be your Henry G. baptized March 24 1849? I didn't find a record for Patrick or William but you may want to scan the parish register https://registers.nli.ie/parishes/1046
I did not find any baptismal records for children of Patrick Cain and Mary Callaghan but likely they were all born before 1838.
I also looked at the 1855 Griffiths Valuation head of household listing for Omey civil parish which includes Clifden https://www.failteromhat.com/griffiths/galway/omey.htm According to your notes Patrick Cain died in 1856 so possibly he was listed in the Griffiths. There is a Patrick Coyne in Moorneen and also a Patrick Kane in Mauneed (Streamstown). No way to know if either is a record for your ancestor.
Let me know if you have any questions.
Name:Gregory KaneDate of Birth:
Date of Baptism:01-Mar-1849Address:Not RecordedParish/District:CLIFDENGender:MaleCountyCo. Galway
Father:Michael KaneMother:Barbara MannionOccupation:
Sponsor 1 /
Informant 1:Valentine MannionSponsor 2 /
Informant 2:B Connela