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Interestingly, someone else posted a question about this exact same family in 2022! They must be a cousin of yours:
https://www.rootschat.com/forum/index.php?topic=859219.0A few good links to the various civil registration records, so worth checking out the thread.
Ancestry has what seems to be some sort of application form for an extract of a parish register, which has the following details:
Full name of applicant: William or William Gilmore Cassidy (Boveedy)
Address: Mr. William Cassidy, Upperlands, County Londonderry
Full names of father and mother of applicant: Manus & Jane
Parish church in which applicant was baptised: Tamlight O'Crilly
Year of baptism or birth: 1850-1852
So it looks like William applied for a copy of his baptism entry? Strange that there is such a large gap between William and John, but I cannot find any other children.
I also could not find anything on John James, but hopefully someone else will have some ideas as south-west Ireland is more my specialty.
Regarding Francis Cassidy, I notice that in the 1831 census of Tamlight O'Crilly there is both a Manus Cassidy and Frank Cassidy present. 3 people in Manus' household and 8 in Frank's. I imagine Frank could be Manus' father and perhaps Manus is Frank's brother or maybe even Frank's father?
Queenie