Dog possibly swallows small bug (4)
I believe the answer is:
pest
'bug' is the definition.
(pest can mean an insect)
'dog possibly swallows small' is the wordplay.
'dog possibly' becomes 'pet' (dogs are often kept as pets).
'swallows' indicates putting letters inside.
'small' becomes 's' (abbreviation - e.g. clothes size).
'pet' going around 's' is 'PEST'.
(Other definitions for pest that I've seen before include "Annoying creature" , "Destructive insect or nuisance" , "Plague" , "Annoyance" , "Destructive creature" .)