NB this is not writ in stone, just possibilities that I see in the records. Without seeing certificates / parish records I can't be sure, but think its worth investigating
looking at the list of children on that 1871 census ShaunJ found for you all of whom are said to have been born in Lanchester
Piece 4956, folio 49, page 10 and seeing what is in the GRO births to try to ascertain a MMN
Durham Reg District included Kyo (in Lanchester Parish) and a blank MMN indicates a single mother
Sarah , born c1855 - no likely Sarah Errington, but there is a Sarah Smith, MMN blank Q1 1855 Durham
Henry, born c1857, no likely Henry Errington, but there is a Henry Smith, MMN blank Q4 1857 Durham
Elizabeth, born c1863, a possible in Durham for Errington Q2 mmn Elliott plus one possible for Smith MMN blank Q3 1863
John born c1865,2 possibles for John Errington, Q3 1865 Durham MMN Watson, Q4 1865 Durham MMN Smith and 1 John Smith, MMN blank Q2 1865
Jane A, born c 1868, Jane Ann Errington, Q2 1868 Durham MMN Smith
Isabella b c1870, FreeBMD has an Isabella Errington Q2 1870 Durham, MMN Smith, GRO has no first name for this child, but has a MMN of Smith and Durham Registrars births search has a Baby Girl Errington in 1870
Difficult to say but possibilities are
they married c 1863-1868
didn't marry at all and whoever registered the later births either fibbed and said they were married or weren't aware they weren't
taking the first possibility & looking for marriages for a John Errington to a lady called Smith 1863 - 1871
the only possible I can find (but others may have more success) is
Q1 1865, Gateshead John Errington and Jane Smith on the same page in the register
and there is a death for a Jane Errington in Q2 1870, Durham, age 36