There's a clue on the map in that it says 'Wirral Railway Slopes Branch'. The Slopes Branch, which opened in 1906, left the main line to Seacombe west of Breck Road Bridge and ran past the bottom of the grounds of Darleydene/The Slopes (hence the name) to join the Dock Board railway at a level crossing on Poulton Bridge Road. The Branch was finally closed in the late 50s when rail traffic to the north side of the Docks declined. The first two photos below show the gated level crossing in 1958, together with the Dock Railway branch going to Bidston Dock.
Getting back to the original topic of this post, the third pic shows the terrace of houses visible in the original postcard - Breck Place - viewed from the Breck Road end in 1961.