Staff harbouring a fugitive? That’s cuckoo! (10)
I believe the answer is:
roadrunner
'cuckoo' is the definition.
(roadrunner is a kind of cuckoo)
'staff harbouring a fugitive?' is the wordplay.
'staff' becomes 'rod' (both can mean a stick or cane).
'harbouring' means one lot of letters goes inside another.
'fugitive?' becomes 'runner' (someone on the run).
'rod' enclosing 'a' is 'road'.
'road'+'runner'='ROADRUNNER'
'that's' acts as a link.
(Other definitions for roadrunner that I've seen before include "Chaparral cock" , "Bird" .)