Complaint - it's angrily dealt with (10)
I believe the answer is:
laryngitis
'complaint' is the definition.
(laryngitis is a kind of health problem)
'it's angrily dealt with' is the wordplay.
'dealt with' is an anagram indicator.
'its'+'angrily'='itsangrily'
'itsangrily' with letters rearranged gives 'LARYNGITIS'.
(Other definitions for laryngitis that I've seen before include "Inflammation of the windpipe" , "Sore throat" , "Inflammation of part of throat" , "Throat infection" , "Inflammation making it difficult to speak" .)