Burns: small rivers flowing to the west, not east (6)
I believe the answer is:
smarts
'burns' is the definition.
'small rivers flowing to the west not east' is the wordplay.
I cannot quite see how this works, but
'small' could be 'sm' and 'sm' is located in the answer.
The remaining letters 'arts' is a valid word which might be clued in a way I don't understand.
This explanation may well be incorrect...
(Other definitions for smarts that I've seen before include "that Sting does" , "is painful" , "Stings, sounding clever" , "Stings or rankles" , "Stings painfully" .)