Maps from 1895, 1930 and 1954 all show it as a refuge siding. Probably chosen as the best place to put away a goods off either line that that might get in the way at St John’s.
It was a carriage siding for much of its life, not sure when it ceased to be used but pre-Networker.
New Beckenham itself had a centre road (which I assumed the OP was asking about!) - I believe it was used to allow up trains to be combined off both routes.
Confirmed by Hajducki's book: introduced in 1897 when the station was rebuilt, removed after electrification. The space is still there.
I have a vague recollection that the berthing siding was used for the Clock House emergency train (the Chaffinch Brook under that station being extremely prone to flood until about 1960).