Exaggerate being delayed supporting popular head of finance (7)
I believe the answer is:
inflate
'exaggerate' is the definition.
(thesaurus)
'delayed supporting popular head of finance' is the wordplay.
'delayed' becomes 'late' (I've seen this before).
'supporting' means one lot of letters go next to another (in a down clue, some letters go below others).
'popular' becomes 'in'.
'head of' suggests taking the first letters.
The first letter of 'finance' is 'f'.
'in'+'f'='inf'
'late' after 'inf' is 'INFLATE'.
'being' is the link.
(Other definitions for inflate that I've seen before include "Exaggerate or fill with air" , "Blow up like balloon" , "Puff up" , "aggrandize them" , "Enlarge" .)