Set off, having assignation outside school (8)
I believe the answer is:
detonate
'set off' is the definition.
('detonate' can be a synonym of 'set off')
'assignation outside school' is the wordplay.
'assignation' becomes 'date' (I've seen this before).
'outside' is an insertion indicator.
'school' becomes 'eton' (famous UK public school).
'date' placed around 'eton' is 'DETONATE'.
'having' is the link.
(Other definitions for detonate that I've seen before include "let off" , "Set off explosive device" , "Start 15 [EXPLOSION]" , "go off with a bang" , "Discharge" .)