Bird inside of skip found by waif, oddly (4)
I believe the answer is:
kiwi
'bird' is the definition.
(kiwi is a kind of bird)
'inside of skip found by waif oddly' is the wordplay.
'inside of' indicates the central letters.
'found by' is a charade indicator (letters next to each other).
'oddly' indicates alternate letters.
The middle of 'skip' is 'ki'.
The alternating letters of 'waif' are 'wi'.
'ki'+'wi'='KIWI'
(Other definitions for kiwi that I've seen before include "New Zealander's bird" , "N Zealand bird" , "apteryx" , "Someone from Auckland?" , "Hairy fruit with green flesh" .)