School's covering my burn (6)
I believe the answer is:
scorch
'burn' is the definition.
(scorch is a kind of burn)
'school's covering my' is the wordplay.
'school' becomes 'sch' (abbreviation).
'covering' indicates putting letters inside.
'my' becomes 'cor' (both are expressions of surprise).
'sch' going around 'cor' is 'SCORCH'.
(Other definitions for scorch that I've seen before include "Mark by overheating" , "Burn slightly" , "Burn on surface, char" , "limit fire damage to outside" , "Damage with heat" .)