Bird that's right? Stick around! (5)
I believe the answer is:
crane
'bird' is the definition.
(crane is a kind of bird)
'right? stick around' is the wordplay.
'right?' becomes 'r' (abbreviation).
'stick' becomes 'cane' (cane is a kind of stick).
'around' means one lot of letters goes inside another.
'r' placed inside 'cane' is 'CRANE'.
'that's' acts as a link.
(Other definitions for crane that I've seen before include "Lifting machine or long-legged bird" , "Device for picking up" , "Wading bird - lifting device" , "American writer" , "Bird; dockside machine" .)