Shoot sergeant-major in Gulf state, one investigating complaints (9)
I believe the answer is:
ombudsman
'one investigating complaints' is the definition.
(an ombudsman investigates complaints)
'shoot sergeant-major in gulf state' is the wordplay.
'shoot' becomes 'bud' (I've seen this in another clue).
'sergeant major' becomes 'SM' (abbreviation for sergeant major).
'in' means one lot of letters goes inside another.
'gulf state' becomes 'oman' (Oman is a Middle Eastern country).
'bud'+'sm'='budsm'
'budsm' put inside 'oman' is 'OMBUDSMAN'.
(Other definitions for ombudsman that I've seen before include "Public official" , "Complaint investigator" , "one appointed to investigate complaints" , "Government appointee who investigates complaints" , "Maladministration investigator" .)