Youngster -- figure inspiring exasperation primarily (4)
I believe the answer is:
teen
'youngster' is the definition.
(I've seen this before)
'figure inspiring exasperation primarily' is the wordplay.
'figure' becomes 'ten' (I've seen this in other clues**).
'inspiring' indicates putting letters inside.
'primarily' says to take the initial letters.
The first letter of 'exasperation' is 'e'.
'ten' going around 'e' is 'TEEN'.
(Other definitions for teen that I've seen before include "Young person" , "Adolescent (informal)" , "Adolescent, then adult-to-be" , "Youngster over twelve" , "Of 13 to 19 year olds" .)