To annoy old flame eat pears messily (10)
I believe the answer is:
exasperate
'to annoy' is the definition.
(I've seen this before)
'old flame eat pears messily' is the wordplay.
'old flame' becomes 'ex' (a former partner).
'messily' is an anagram indicator.
'eat'+'pears'='eatpears'
'eatpears' anagrammed gives 'asperate'.
'ex'+'asperate'='EXASPERATE'
(Other definitions for exasperate that I've seen before include "bug" , "See axe part (anag.)" , "Annoy" , "Intensely irritate" , "Drive to distraction" .)