Copy owner of tiny hand getting frozen cold (5)
I believe the answer is:
mimic
'copy' is the definition.
(I know that mimic is a type of copy)
'owner of tiny hand getting frozen cold' is the wordplay.
'owner of tiny hand getting frozen' becomes 'mimi' (reference to the opera La bohme).
'cold' becomes 'c' (eg on taps).
'mimi'+'c'='MIMIC'
(Other definitions for mimic that I've seen before include "Impressionist" , "Impersonate" , "Imitate for humorous reasons" , "Imitate, especially for satirical effect" , "Caricature" .)