Nora
The Patrick burial register shows an Anne cain , daughter of Robert died age 3 27 Dec 1840, the only Anne born to a Robert at this time was the one you mentioned- mother Christian Fargher.
I notice in the 1881 census a mother in law Ann Murray was living with John and Anne wich means Anne was either illegitimate or her mother married again. George Murray married Ann cain 1 June 1844 and had a son George who died after 6 weeks 18 May 1845. George ,himself, died 9 Jan 1866 at Knockaloe.
It is likely that your Anne Cain was the daughter of John Cain and Ann Quayle ch. 26 Aug 1832 in Patrick and John would have died between his last born Elizabeth and the marriage of the Murrays. A John Caine died age 30 17 July 1834.

Len Faragher