Poet's put up in islands belonging to Mediterranean state (7)
I believe the answer is:
israeli
'belonging to mediterranean state' is the definition.
(belonging to Israel)
I cannot understand how the remainder of the clue works.
(Other definitions for israeli that I've seen before include "A Middle-Easterner" , "One from Tel Aviv, perhaps" , "Ali, rise! (anag) - Middle Easterner" , "Citizen of a Middle Eastern nation" , "Jewish character" .)