Pompous up-starts getting under my skin (4)
I believe the answer is:
smug
'pompous' is the definition.
Both the answer and definition are adjectives. Perhaps there's a link between them I don't understand?
'up-starts getting under my skin' is the wordplay.
'up' is a reversal indicator (in down clue: letters go upwards).
'starts' suggests taking the first letters.
The initial letters of 'getting under my skin' is 'gums'.
'gums' backwards is 'SMUG'.
Can you help me to learn more?
(Other definitions for smug that I've seen before include "Prim" , "Overly complacent" , "Be envious of" , "Unctuous" , "Self-satisfied and vain" .)