No half-term upset in school (4)
I believe the answer is:
eton
'school' is the definition.
(famous UK public school)
'no half-term upset' is the wordplay.
'half-term' becomes 'te'.
'upset' says the letters should be written backwards.
'no'+'te'='note'
'note' backwards is 'ETON'.
'in' is the link.
(Other definitions for eton that I've seen before include "Place for learning" , "Kind of collar or jacket (worn at college?)" , "Public school, established in 1440" , "Town on the Thames noted for its school" , "Leading English public school" .)