Staff cross when Oscar is dismissed (3)
I believe the answer is:
rod
'staff' is the definition.
(both can mean a stick or cane)
I cannot understand how the rest of the clue works.
(Other definitions for rod that I've seen before include "Old saying 'Spare the . . . and spoil the child'" , "Wand - fishing stick" , "A few yards" , "Long, straight stick" , "Flexible stick for 11 Down" .)