A cold man having performed (5)
I believe the answer is:
acted
'performed' is the definition.
(acting is a kind of performing)
'a cold man' is the wordplay.
'cold' becomes 'c' (abbreviation, eg on taps).
'man' becomes 'ted' (Ted is a man's name).
'a'+'c'+'ted'='ACTED'
'having' is the link.
(Other definitions for acted that I've seen before include "Performed on stage" , "Pretended" , "appeared on stage" , "Behaved or played a part" , "took part in simulation" .)