Singe top of branch and run off (4)
I believe the answer is:
burn
'singe' is the definition.
(thesaurus)
'top of branch and run off' is the wordplay.
'top of' says to take the initial letters.
'and' says to put letters next to each other.
'off' is an anagram indicator.
The initial letter of 'branch' is 'b'.
'run' anagrammed gives 'urn'.
'b'+'urn'='BURN'
(Other definitions for burn that I've seen before include "Small stream in Scotland" , "Scorch; brook" , "Catch light; stream" , "runner?" , "Scottish brook" .)