Hello Lleucu,
Have you seen the entry in the parish registers for their marriage in 1834? This might reveal the name and occupation of Mary's father.
I also have ancestors who married in Liverpool. They were from Mold, Flintshire and Chester/Cheshire. I don't know of any obvious connection to Liverpool. Why they should marry in Liverpool is a mystery to me. I do wonder if the marriage was disapproved of by one, or both, sets of parents and they went out of the area. One possibility is that the groom was working for the railway and working in Liverpool when they married.
The 1841 census shows Matthew and Elizabeth Clifford living in Pulford village. The household includes Sarah Clifford, age 20 and James Clifford, age 15. All four people show they were born in county, ie Cheshire. Relationships were not given in the 1841 but it looks reasonable to assume that Sarah and James were children of Matthew and Elizabeth Clifford. Bear in mind that in the 1841 census, ages were rounded up, or down, to the nearest 5 years, so 20 and 15 would be rounded figures.
I don't know if you have tried to link Sarah and James to Mary Clifford. I noticed that Sarah married in St Oswald's, Chester, in 1843. Her father is shown as Matthew Clifford.
I noticed that Matthew Clifford was buried in Pulford in 1860. Betty was buried there in 1865.
Elizabeth Clifford can be found in the 1861 census in Moor Bridge, Saltney. She is aged 81, a widow and described as "Mother". The head of the household is William Evans, age 40, Railway Lab. born Pulford, his wife is Martha Evans, age 42, born Pulford.
I wonder if Elizabeth was married before in view of her age difference to Matthew Clifford. Could Mary have been a daughter by her first husband?
My advice would be to put a post on the Cheshire Board requesting that someone do a look up in Chester RO to see if they can find a baptism for Mary Clifford in Pulford. It could be that she was baptised in one of the churches in Chester however.
Jo.