Bear periodically brash pompous official (4-3)
I believe the answer is:
pooh-bah
'pompous official' is the definition.
(I've seen this before)
'bear periodically brash' is the wordplay.
'bear' becomes 'pooh' (Winnie the Pooh).
'periodically' means one should take alternating letters (letters must be selected 'every so often').
The alternating letters of 'brash' are 'bah'.
'pooh'+'bah'='POOH-BAH'
(Other definitions for pooh-bah that I've seen before include "Arrogant person of importance" , "Pompous official" , "Pompous office-holder" .)