The A2310 number has been allocated by the A-Z Street Atlas to a short link road in Chatham, connecting the A231 with the A289 along Dock Road and Maritime Way, past the Historic Dockyard.
It seems that the number is either a mistake or a copyright trap. Other maps, for example the Ordnance Survey, show the road to be unclassified and the road number does not appear on any signs. The official list of classified roads published by Medway Council gives the road the number C372.