LSE admitting a professor’s first mistake (5)
I believe the answer is:
lapse
'mistake' is the definition.
(lapse is a kind of mistake)
'lse admitting a professor's first' is the wordplay.
'admitting' means one lot of letters goes inside another.
'professor's first' becomes 'p' (1st letter of 'professor').
'a'+'p'='ap'
'lse' going around 'ap' is 'LAPSE'.
(Other definitions for lapse that I've seen before include "Pales into omission" , "Momentary mistake" , "Revert (to bad habits, etc)" , "moral shortcoming" , "breach of protocol?" .)