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So can we assume, so far we have:
James Kennedy (b circa 1851) married one Mary SMITH in 1870 and had sons John (1874) and James (1876)
Then with perhaps something happening to his wife, he remarried in 1878 Ellen MCGR(E)ADY and they go on to have John (1877 - a year before their marriage though?) Daniel, Catherine, Stephen and Peter?
Not if this is Mary (nee Smith) and James Kennedy in the 1881 census which I posted earlier:
https://www.familysearch.org/ark:/61903/1:1:Q27L-M86QPossible marriage:
Marriages Sep 1874
Kennedy James Liverpool 8b 453
Smith Mary Ann Liverpool 8b 453
Childrens birth registrations:
KENNEDY, JAMES SMITH
GRO Reference: 1876 S Quarter in LIVERPOOL Volume 08B Page 6
KENNEDY, CHARLES SMITH
GRO Reference: 1878 D Quarter in LIVERPOOL Volume 08B Page 17
KENNEDY, WILLIAM SMITH
GRO Reference: 1880 D Quarter in LIVERPOOL Volume 08B Page 57
The James Kennedy son of James and Ellen in this list was born in 1878
Census 1911, at 15 Dryen St (same as Daniel KENNEDY with the niece and nephew) is James KENNEDY, wife Mary and three sons. Also, Peter KENNEDY, 22 years, boarder, all born Liverpool.
KENNEDY, John mms. MCGRADY
1877 JQ LIVERPOOL Vol 08b pg 28
KENNEDY, Daniel mms. MCGREADY
1882 JQ LIVERPOOL Vol 08b Page 65
KENNEDY, Catherine Christina MCGRADY
1885 MQ LIVERPOOL Vol 08b pg 24
KENNEDY, Stephen mms. MCGRADY
1886 SQ LIVERPOOL Vol 08b pg 60
KENNEDY, Peter mms. MCGRADY
1888 SQ Liverpool Vol 08b pge 38
FreeBDM JQ 1878 Liverpool Vol 8b pg410
KENNEDY James m. Ellen McGRADYperhaps your James Kennedy was a different one to the one who married Mary Smith and your James Jnr was a little younger than you thought
Edit: Sorry just noticed that's a marriage for James and Ellen in 1878 not a birth for a James