Odd characters in secret exercises on the staff (7)
I believe the answer is:
sceptre
'the staff' is the definition.
(sceptre is a kind of staff)
'odd characters in secret exercises on' is the wordplay.
'odd characters in' means one should take alternating letters (the letters at odd-numbered positions).
'exercises' becomes 'PT' (Physical Training).
'on' becomes 're' ('on' can be mean 'with reference to').
The alternating letters of 'secret' are 'sce'.
'sce'+'pt'+'re'='SCEPTRE'
(Other definitions for sceptre that I've seen before include "Royal symbol" , "Official staff" , "Royal baton" , "Royal staff (paired with 2 down)" , "Rod of office" .)