Express and Sun primarily showing arrogance (4)
I believe the answer is:
airs
'arrogance' is the definition.
(to 'put on airs' is to behave arrogantly)
'express and sun primarily' is the wordplay.
'express' becomes 'air' (I've seen this before).
'and' means one lot of letters go next to another.
'sun primarily' becomes 's' (abbreviation. I am not sure about the 'primarily' bit.).
'air'+'s'='AIRS'
'showing' acts as a link.
Can you help me to learn more?
(Other definitions for airs that I've seen before include "Tunes and affectations" , "Ventilates or tunes" , "feelings" , "- - and graces" , "Pretentiousness" .)