LSE admitting reactionary old man’s mistake (5)
I believe the answer is:
lapse
'mistake' is the definition.
(lapse is a kind of mistake)
'lse admitting reactionary old man's' is the wordplay.
'admitting' indicates putting letters inside.
'reactionary' says the letters should be written in reverse (reactions happen in the opposite direction).
'old man' becomes 'pa' (both can mean father).
'pa' back-to-front is 'ap'.
'lse' going around 'ap' is 'LAPSE'.
(Other definitions for lapse that I've seen before include "Slip back" , "Interval" , "Careless mistake" , "Brief (memory) failure" , "Pleas about omission" .)