Wrongdoing of troublesome brat the head expelled (5)
I believe the answer is:
error
'wrongdoing' is the definition.
(synonyms)
'troublesome brat the head expelled' is the wordplay.
'troublesome brat' becomes 'terror' (term for a misbehaving child).
'the head expelled' means to remove the first letter (the head of the word is removed).
'terror' with its first letter removed is 'ERROR'.
'of' acts as a link.
(Other definitions for error that I've seen before include "Solecism" , "Faulty judgment" , "Boo-boo" , "Faux pas" , "wrong" .)