Muddy deposit in British river, we hear (4)
I believe the answer is:
ooze
'muddy deposit' is the definition.
(mud at the bottom of a river)
'british river we hear' is the wordplay.
'british river' becomes 'Ouse' (name of several British rivers).
'we hear' indicates a 'sounds like' (homophone) clue.
'ouse' is a homophone of 'OOZE'.
'in' acts as a link.
(Other definitions for ooze that I've seen before include "Percolate" , "Exude mud" , "Seep, exude" , "Trickle slowly out, like mud" , "Emit" .)