Dog perhaps that's bitten son is an irritation (4)
I believe the answer is:
pest
'an irritation' is the definition.
Although both the answer and definition are singular nouns, I can't understand how they can define each other.
'dog perhaps that's bitten son' is the wordplay.
'dog perhaps' becomes 'pet' (dogs are often kept as pets).
'that's bitten' is an insertion indicator (inserted letters bite around others).
'son' becomes 's' (genealogical abbreviation for son).
'pet' going around 's' is 'PEST'.
'is' acts as a link.
Can you help me to learn more?
(Other definitions for pest that I've seen before include "Plague" , "Annoyance" , "Annoying thing" , "Nuisance; half a European capital" , "Troublesome person" .)