Jean, following up on your Robert Cain who was a carter in 1881, 1891 and 1901 (unmarried), and your idea that John might have used his mother's surname, I had a look at Braddan bapts for illegitimate births. Braddan because Liz wrote that John was born in Union Mills.
These are extracted bapts on the IGI:
JOHN CORLETT Christening: 16 AUG 1868 Braddan, Isle Of Man
Mother: CATHE CORLETT [married JAMES CORKILL 26 MAR 1870 Braddan?]
JOHN CLUCAS Christening: 28 FEB 1862 Braddan, Isle Of Man
Mother: ANN CLUCAS [married WM. CANNELL 06 SEP 1868 Braddan?]
These are some Cains. I wondered whether it was possible that Robert was John Robert?
JOHN HENRY CAIN Christening: 06 JUN 1870 Braddan, Isle Of Man
Father: JOHN CAIN Mother: JANE REDPATH
JOHN HENRY CAIN Christening: 09 OCT 1870 Braddan, Isle Of Man
Father: JOHN CAIN Mother: JANE REDPATH
[These two are odd - both extracted entries.]
JOHN CAIN Christening: 19 JUL 1868 Braddan, Isle Of Man
Father: JOHN CAIN Mother: CATHE CORKILL
Sue