Nip inside branch for business chat (4,4)
I believe the answer is:
shop talk
'business chat' is the definition.
'nip inside branch' is the wordplay.
'nip' becomes 'hop' (both can mean to make a short visit).
'inside' indicates putting letters inside.
'branch' becomes 'stalk'.
'hop' inserted inside 'stalk' is 'SHOP TALK'.
'for' is the link.
(Other definitions for shop talk that I've seen before include "business, as discussed elsewhere" , "Chat about business especially out of hours" , "chat about job to be done?" , "business jargon" , "after-hours conversation?" .)