Deputy to contend with cold (4)
I believe the answer is:
vice
'deputy' is the definition.
(as in vice-president)
'contend with cold' is the wordplay.
'contend' becomes 'vie' (synonyms).
'with' means one lot of letters goes inside another.
'cold' becomes 'C' (eg on taps).
'vie' enclosing 'c' is 'VICE'.
'to' acts as a link.
(Other definitions for vice that I've seen before include "Strong gripper" , "Workshop tool - bad habit" , "Gripper -- fault" , "Gripping tool - bad habit" , "Is the gripping tool evil?" .)