Gulf state picked up ENT problem (7)
I believe the answer is:
catarrh
'ent problem' is the definition.
(problem for ear, nose and throat department)
'gulf state picked up' is the wordplay.
'gulf state' becomes 'qatar' (I've seen this before).
'picked up' shows a homophone (sound like).
'qatar' sounds like 'CATARRH'.
(Other definitions for catarrh that I've seen before include "Problem with cold" , "Arch rat with inflamed mucous membrane" , "Nasal problem" , "Inflammation of membrane of nose and throat" , "Inflammation of mucous membrane with watery discharge" .)