Month not the first that may be triumphal (4)
I believe the answer is:
arch
'triumphal' is the definition.
Both the answer and definition are adjectives. Maybe you can see a link between them that I can't see?
'month not the first' is the wordplay.
'month' becomes 'march' (March is an example).
'not the first' means to remove the first letter.
'march' with its first letter taken away is 'ARCH'.
'that may be' acts as a link.
Can you help me to learn more?
(Other definitions for arch that I've seen before include "in short, is out of date" , "Curve spanning an opening" , "part of fingerprint" , "Curved overhead structure" , "Bend into upward curve" .)