Onset of whooping in sickly bird (7)
I believe the answer is:
swallow
'bird' is the definition.
(swallow is a kind of bird)
'onset of whooping in sickly' is the wordplay.
'onset of' indicates taking the first letters.
'in' means one lot of letters goes inside another.
'sickly' becomes 'sallow' ('sallow' can be a synonym of 'sickly').
The initial letter of 'whooping' is 'w'.
'w' put into 'sallow' is 'SWALLOW'.
(Other definitions for swallow that I've seen before include "Get down" , "One doesn't make a summer, they say" , "Fall for; bird" , "Songbird" , "Bird - gulp down" .)