Conceited person's report covered by flier (8)
I believe the answer is:
popinjay
'conceited person's' is the definition.
(I've seen this before)
'report covered by flier' is the wordplay.
'report' becomes 'pop' ('report' can mean a sudden explosive noise).
'covered' becomes 'in' (I've seen this in other clues).
'by' means one lot of letters go next to another.
'flier' becomes 'jay' (jay is a kind of birdanimal**).
'pop'+'in'+'jay'='POPINJAY'
(Other definitions for popinjay that I've seen before include "One super-smart" , "Strutting fop" , "vain sort" , "Conceited person" , "Foppish type" .)