Periods of greater aspiration (4)
I believe the answer is:
eras
'periods' is the definition.
(periods of time)
'of greater aspiration' is the wordplay.
'of' says the answer is hidden in the clue.
'ERAS' is hidden in the letters of 'greater aspiration'.
(Other definitions for eras that I've seen before include "Periods of time" , "typeface" , "Periods, ages" , "At least two times" , "Long and distinct periods of history" .)