One leaves the old governor's band (5)
I believe the answer is:
Here is my best explanation:
'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 "Fastening strip", "Secure", "beat", "Elongated leather strip", "tie", "Thong", "something to hang on to", "punishment in school", "Parts (anag.)" and "Parts of a leather band".)