I believe this is them in 1861 - the Enumerator has mistranscribed the information from the original census sheet badly (i.e. he has Michael down as John). I have seen occurences of this before.
1861 England Census- Jarrow, South Shields (registration district), Hedworth, Monkton and Jarrow (civil parish), Co.Durham
John Wood 28
Elizabeth Wood, wife, 27, born Ireland
Mary A Wood 3 b. Jarrow
Alice Wood 2 b. Jarrow
John English 25, Brother in law b. Ireland
Edward Murray 48 Boarder, Labourer, b. Ireland
John must have gone back to Ireland that year and married Catherine. Possibly Michael's sister?
You probably have them in 1871 but here it is just in case :
1871 England Census- Liverpool, Toxteth Park (civil parish), St. James (ecclesiastical parish), West Derby (registration district), Lancashire
Michael Woods, 36, born Ireland
wife Elizabeth Woods
Alice Woods 11
John Woods 9
Elizabeth Woods 7
Catherine Woods 5
Michael Woods 2
Harriet Woods 5 Months
William Mottram 63
Elizabeth Dougherty 25