Sulk, extremely testy and small-minded (5)
I believe the answer is:
petty
'small-minded' is the definition.
('petty' can be a synonym of 'small-minded')
'sulk extremely testy' is the wordplay.
'sulk' becomes 'pet'.
'extremely' suggests removing the centre.
'testy' with its middle taken out is 'ty'.
'pet'+'ty'='PETTY'
'and' is the link.
(Other definitions for petty that I've seen before include "Insignificant and smallminded" , "Insignificant or trivial" , "Small-time" , "Inferior" , "Trivial, piffling" .)