In 1857S, the GRO marriage indices have a Thirza (or Thurza) Broad marrying William Harper in Wantage. This is the only record I have found that refers to William.
Thurza Broad appears in the 1841 (age 5) and 1851 (aged 19) censuses with her parents Henry and Amy in Wantage, Berkshire.
Thurza appears in the 1861 census with her parents in Wantage as 30 year old unmarried (not widowed) Thurza Harper with a daughter Louisa aged 1. (Louisa Harper's birth registration took place in 1859D in Wantage with the mother's maiden name blank; this is the only birth registration with the right name in the right district around the right time).
Thurza goes on to have 3 further children: Frederic William Harper (1863 Wantage); Michael Joseph Harper (1865 Wantage); and Mary Jane Harper (1867 Highworth, Wiltshire). All these children are registered with a blank mother's maiden name. (I've identified the siblings by finding Louisa in the 1871 census and tracing the birth registrations of the siblings with her).
Thurza dies in 1867S in Abingdon. In 1871, her children are all in the Wantage Workhouse; in 1881, the two youngest children are still there.
I cannot find a likely death for William Harper, the putative father.
My question is: what scenarios could result in the children being registered with the surname Harper and no mother's maiden name?