Can't see a child living with them in 1861 or 1871 but in 1881 was this the family with son John W - born pre the 1855 marriage - living at Annfield Plain??
John Errington   56 Colliery Overman (Miner) b Kyo, Durham
Cath. Errington   49 b S Shields, Durham,
John W. Errington 28 Coal Miner b Kyo, Durham
Possibly birth pre marriage
ANDERSON, JOHN  WILLIAM      -No mothers maiden name    
GRO Reference: 1852  J Quarter in DURHAM  Volume 10A  Page 148 
And in 1861 and 1871 living as John W Anderson with grandparents at West Kyo
1871 
William Anderson   75  Time Keeper at Coke Works b Scotland
Mary Anderson   71 b Jesmond, Northumberland
John W Anderson   19 grandson Coal Miner b Kyo