Farm centre managed on a Scottish island (5)
I believe the answer is:
arran
'a scottish island' is the definition.
(island off the coast of Scotland)
'farm centre managed on' is the wordplay.
'centre' says to take the centre.
'managed' becomes 'ran' (both can mean organised or was in charge of).
'on' is a charade indicator (letters next to each other).
The centre of 'farm' is 'ar'.
'ar'+'ran'='ARRAN'
(Other definitions for arran that I've seen before include "Scottish territory" , "Isle near Firth of Clyde" , "Firth of Clyde island" , "Scottish island" .)