Bug of irritating nature (4)
I believe the answer is:
gnat
'bug' is the definition.
'of irritating nature' is the wordplay.
'of' indicates the answer is hidden within the clue.
'GNAT' can be found hidden inside 'irritating nature'.
(Other definitions for gnat that I've seen before include "Small bloodsucker" , "Small (swarming) fly" , "Stinging insect" , "Two-winged fly" , "Something with bite" .)