Not sure if you already have this information, but Joseph Clarke married Ann King, as you say, on 6.7.1829 at St Marys, Nottingham.
I did notice that the spelling of Salome's surname was Clark not Clarke, don't know if that makes a difference when searching.
On the 1851 census I've come across the following:-
Ann Clark aged 42, widowed, living with her 2 daughters at 8 Denmark Court, Nottingham. They were all working in the textile industry. The daughters were recorded as:-
Solome Asnith aged 20 unmarried (it says on the transcript that it could actually be Clarke and the other daughter - Joseph Ann Clark aged 18!
On the 1861 census I have found an Ann Clark aged 52, widowed, living with one daughter Salina, aged 31, occupation milliner and smallware dealer. They are living at 3, Hollowstone, St Marys Nottingham.
1871 they are living in Liverpool, mother aged 62, Salome A Clarke, aged 41, milliners,
on the 1881 census they are still in Liverpool, both still milliners, mother aged 72, daughter 51. Salome's surname is given as either Clarke (with an E) or Monath on this census.
I've found a marriage for a Joseph Ann Clark - 13th Nov 1853, she married a Samuel Burton, at St Marys Nottingham.
That's all I've managed to find, hopefully it'll be of some help in filling in the gaps. If I find anything else I'll let you know.
BG