I think this is your Elizabeth:

1851 census:
Reeves Yard, Barter(?) St, Liverpool [HO 107/2181 f.124 p.20 sch.63]
Benjamin Mosson - head - mar - 40 - Dock labourer - born Wales, Wrexham
Elizabeth " - wife - mar - 38 - born Isle of Man
Thomas " - son - 15 Shipwright (apt) - Lancs Liverpool
Eliza " - dau - 13 - dressmaker - Liverpool
Sarah Ann " - dau - 10 - scholar - Liverpool
Jane Moore - servt - un - 48 - general servant - Isle of Man

The way double 's' was sometimes written can indeed look like a 'p'. I searched on Ancestry with 'Sarah Ann born 1841 mother Elizabeth'.

I see in the 1871 census the daughter's married name is Anderson:

freebmd has Sarah A Mosson m. Christian Anderson last quarter 1863, 8b 278 if you want the cert, which would give father's name etc.

By the way, the 1871 census does read Mosson, though I can understand how it could be read Morson, and Elizabeth's age looks more like 69, although not very clear. Couldn't spot them in a quick search of 1861.

Hope that helps.