Lothario's shrewder when touring gulf state (9)
I believe the answer is:
womaniser
'lothario's' is the definition.
(I've seen this before)
'shrewder when touring gulf state' is the wordplay.
'shrewder' becomes 'wiser' (I can't justify this - if you can you should give a lot more credence to this answer).
'when touring' indicates putting letters inside.
'gulf state' becomes 'oman' (Oman is a Middle Eastern country).
'wiser' going around 'oman' is 'WOMANISER'.
Can you help me to learn more?
(Other definitions for womaniser that I've seen before include "A rake" , "Philanderer" , "Casanova" , "Playboy" , "Ladies' man" .)