Hesitation by private investigator regarding judge (6)
I believe the answer is:
umpire
'judge' is the definition.
(I know that umpire is a more specific form of the action judge)
'hesitation by private investigator regarding' is the wordplay.
'hesitation' becomes 'um' (sound made when hesitating).
'by' means one lot of letters go next to another.
'private investigator' becomes 'PI' (abbreviation).
'regarding' becomes 're' (re can mean regarding, concerning).
'um'+'pi'+'re'='UMPIRE'
(Other definitions for umpire that I've seen before include "Arbiter of cricket match" , "Kind of referee as in cricket" , "One enforcing the rules in games" , "Sports arbiter" , "The ref in cricket match" .)