An insect's awfully persistent (9)
I believe the answer is:
incessant
'persistent' is the definition.
(I've seen this before)
'an insect's awfully' is the wordplay.
'awfully' indicates anagramming the letters.
'an'+'insects'='aninsects'
'aninsects' is an anagram of 'INCESSANT'.
(Other definitions for incessant that I've seen before include "Continuous" , "Instances (anag)" , "Unending" , "without a break" , "Never-ending" .)