Alfred Kitching (1808 - 1882), a Quaker industrialist, was a partner with his brother William, in W. & A. Kitching at the Hope Town Foundry in Darlington where locomotives were built for the Stockton and Darlington Railway including engines designed by Timothy Hackworth. In 1845 he bought out his brother and operated as A. Kitching. In 1860, The S.&D.R. took their maintenance in-house and purchased the Hope Town Foundry. The remaining parts of the business passed to Kitching's cousin, Charles I'Anson who moved to a foundry at Whessoe, the Whessoe company still operates today. Alfred Kitching became mayor of Darlington in 1870.
It is not known whether the engine numbers relate to engines previously built or if the designs are for engines being built at the time and numbering started again with the new company (note how 19 and 20 are S.&D.R. 58 & 59). Tomlinson discusses a 'Hackworth' engine built in 1851. Ahrons describes and illustrates a number of engines built by both Kitching companies, including the 'Derwent' class built from 1845 to 1848 and the 'Woodlands' built in 1848 for the S.&D.R. A more extensive list of engines built is provided by Grace's Guide - https://www.gracesguide.co.uk/W._and_A._Kitching
(Tomlinson, W.W., The North Eastern Railway, Its Rise and Development [1914])
(Ahrons, E.L., The British Steam Railway Locomotive, 1825-1925, [1927])