Turns up, it's wrong according to the judge to run (4)
I believe the answer is:
trot
'run' is the definition.
(trotting is a kind of running)
'turns up it's wrong' is the wordplay.
'turns up' says the letters should be written backwards.
'it's wrong' becomes 'tort' (legal term).
'tort' written backwards gives 'TROT'.
'according to the judge to' acts as a link.
This may not be right. Some or all of it may belong to another bit of the clue.
Can you help me to learn more?
(Other definitions for trot that I've seen before include "Not dead" , "Horse's pace - Communist" , "Move like a horse" , "Left wing" , "An elderly woman" .)