Explode one day outside public school (8)
I believe the answer is:
detonate
'explode' is the definition.
('detonate' can be a synonym of 'explode')
'one day outside public school' is the wordplay.
'one day' becomes 'date' (date is a kind of day**).
'outside' means one lot of letters goes inside another.
'public school' becomes 'eton' (I've seen this before).
'date' placed around 'eton' is 'DETONATE'.
(Other definitions for detonate that I've seen before include "Discharge" , "Trigger" , "let off" , "Set off explosive device" , "Cause to explode" .)