Privately owned island in clear-up after resettlement (8)
I believe the answer is:
peculiar
'privately owned' is the definition.
(peculiar can mean belonging to or associated with)
'island in clear-up after resettlement' is the wordplay.
'island' becomes 'i'.
'in' is an insertion indicator.
'after resettlement' indicates an anagram.
'clear'+'up'='clearup'
'clearup' with letters rearranged gives 'pecular'.
'i' put inside 'pecular' is 'PECULIAR'.
(Other definitions for peculiar that I've seen before include "Unusual" , "'Strange, odd (8)'" , "Odd - individual" , "I clear up, being odd" , "Extraordinary" .)