Little time to avoid damage to reputation for Greek character (5)
I believe the answer is:
sigma
'greek character' is the definition.
(letter from the Greek alphabet)
'little time to avoid damage to reputation' is the wordplay.
'little time' becomes 't' (abbreviation for time).
'to avoid' is a deletion indicator.
'damage to reputation' becomes 'stigma'.
'stigma' with 't' taken out is 'SIGMA'.
'for' is the link.
(Other definitions for sigma that I've seen before include "Greek letter used as summation sign" , "letter to Greeks" , "Allowing no relaxation" , "The 18th letter of the Greek alphabet" , "character in Corfu" .)