So far it looks as though this is what you have-
William Black married (1852) Fanny Anderson
1. Harriet (9 in 1861)
2. Luke (6 in 1861)
3. Martha (3 in 1861)
4. Sarah M. (1 in 1861)
However, the facts don't quite agree with this-
Harriet's birth gives mother's maiden name as Willis
Luke's birth gives mother's maiden name as Willis
Martha's birth gives mother's maiden name as Willis
Sarah Mary's birth gives mother's name as Willis
(all checkable on GRO's site)
Then we have son daughter born 1870 in Ireland giving mother's maiden name as Willis.
All these records can't be wrong. Therefore, either it is not the correct family or Fanny's maiden name is Willis. It's possible that
1) Fanny was Willis and married to a Mr. Anderson before she married William Black or
2) she went by both surnames (illegitimate, stepfather's name or name of mother's partner. etc.)
You really do need to get the marriage certificate to confirm Fanny's details.