Reportedly go stealthily after a wading bird (5)
I believe the answer is:
stork
'a wading bird' is the definition.
(stork is a kind of wading bird)
'reportedly go stealthily after' is the wordplay.
'reportedly' shows a homophone (sound like).
'go stealthily after' becomes 'stalk' (to follow someone stealthily).
'stalk' is a homophone of 'STORK'.
(Other definitions for stork that I've seen before include "Baby-carrying bird?" , "Mythological baby-bringer" , "Bird that delivers babies?" , "The baby bird, so to speak" , "Large wading bird" .)