Hi Zephyr,
I eventually found a family which seems to fit, but they are transcribed as "Halker", and I must admit, looking at the census entry, it does look like Halker....anyway...
1871 - living at Dales Court, St. Crux, York - ref RG10/4751 Folio 12 Page 13
Joseph Halker, 42, Coal Dealer, born York
Mary wife, 42 born York
George son 22 Striker born York
Mary Ann dau 17 "
Sarah dau 13 "
Joseph son 10 "
John son 3 "
There is a family living at the same address in 1881, this time indexed and looking like Walker - remaining children's names and ages about right, so I'm guessing the 1871 must be right too....we hope....
1881, living at Dales Court, York, Ref RG11/4724 Folio 10 Page 14
Joseph Walker, 53, Labourer,
Mary, 49
Joseph 18
John 13 - all born York.
Hope it helps,
Yorkslass