Go off having assignation outside school (8)
I believe the answer is:
detonate
'go off' is the definition.
(I know that go off can be written as detonate)
'assignation outside school' is the wordplay.
'assignation' becomes 'date' (I've seen this before).
'outside' indicates putting letters inside.
'school' becomes 'eton' (famous UK public school).
'date' enclosing 'eton' is 'DETONATE'.
'having' is the link.
(Other definitions for detonate that I've seen before include "Discharge" , "Trigger" , "Cause to explode" , "Blow up, explode" , "Start 15 [EXPLOSION]" .)