Beat record after a touch of finesse (4)
I believe the answer is:
flog
'beat' is the definition.
(flogging is a kind of beating)
'record after a touch of finesse' is the wordplay.
'record' becomes 'log' (logging is a kind of recording).
'after' is a charade indicator (letters next to each other).
'a touch of' says to take the initial letters.
The first letter of 'finesse' is 'f'.
'log' after 'f' is 'FLOG'.
(Other definitions for flog that I've seen before include "Thrash" , "Swing a cat, perhaps" , "- - a dead horse" , "Hit - sell" , "-- Hitler" .)