One leaves the old governor's band (5)
I believe the answer is:
strap
'band' is the definition.
(strap is a kind of band)
'one leaves the old governor's' is the wordplay.
'one' becomes 'a'.
'leaves' is a deletion indicator.
'the old governor' becomes 'satrap' (I've seen this before).
'satrap' with 'a' taken away is 'STRAP'.
(Other definitions for strap that I've seen before include "Belt; leather strip" , "e.g. Belt" , "Length or loop, of leather perhaps" , "punishment in school" , "Leather fastening" .)