Deep river or burn (4)
I believe the answer is:
sear
'burn' is the definition.
(sear can mean to burn)
'deep river' is the wordplay.
'deep' becomes 'sea' (I've seen this before).
'river' becomes 'r' (abbreviation for river).
'sea'+'r'='SEAR'
'or' is the link.
(Other definitions for sear that I've seen before include "Burn briefly" , "Withered" , "Cauterise" , "Make very hot and dry" , "Scorch or burn the surface of" .)