To improve best mark immediately, correct an error (3,3,6,8)
I believe the answer is:
set the record straight
'correct an error' is the definition.
I don't know anything about this answer so I can't judge whether it can be defined by this definition.
'to improve best mark immediately' is the wordplay.
I cannot really understand how this works, but
'best' could be 'record' (similar in meaning) and 'record' is present in the answer.
'mark' could be 'set' (synonyms) and 'set' is present in the answer.
'immediately' could be 'straight' (as in straight away) and 'straight' is found in the answer.
The remaining letters 'the' is a valid word which might be clued in a way I don't see.
This explanation may well be incorrect...
Can you help me to learn more?
(Other definitions for set the record straight that I've seen before include "Correct a false impression" , "Make the truth known" , "Put right a false impression" , "correct report" , "Stop skipping, perhaps" .)