James Randall was the father of Catherine Randall.
In the 1871 census there is a James Randall (19, born in Kirdford) living with his widowed father William Randall (54, born in Wisborough Green in Sussex). Both are agricultural labourers, but they have an Ann Durant (56) who is a general servant.
He married Harriet Adsett in Petworth on 12th October 1872. She was born in 1848 in Northchapel, England.
Catherine was born on the 18th November 1873, just over a year after they were married.
By 1891 they had moved to 38 Albion Hill in Brighton. James was 39, and still a police constable, his wife Harriet was 43, and their daughter Catherine was 17 and working in a factory. They also had a boarder (Herbert Upperton, who was a 13 year old baker's assistant)
By 1901 James was 51 and had retired, and was living with Harriet (54) in Rose Cottage, Staplefield Common, Handcross (part of), Cuckfield, Sussex.
In 1911 James (60) is an Innkeeper at the St George Thomas's Inn in Slindon (Sussex), and Harriet (64) is assisting him in the business. Their granddaughter Mabel Richardson is living with them. According to this census they only had one child (and given that Harriet is 64, it is unlikely they would have had any more).
There are two possible marriage entries in the indices for James Randall (Petworth is 5 miles from Kirdford):
James Randall, Harriett Adnettt, 1872, Oct-Nov-Dec, Petworth, Sussex, Vol 2b, Page 557