Habit to drop cold summer's activity (8)
I believe the answer is:
addition
'summer's activity' is the definition.
(doing sums)
'habit to drop cold' is the wordplay.
'habit' becomes 'addiction' (drug addiction).
'to drop' suggests deleting specific letters.
'cold' becomes 'c' (eg on taps).
'addiction' with 'c' removed is 'ADDITION'.
(Other definitions for addition that I've seen before include "Something extra" , "Calculation of the sum of two or more" , "Combination of numbers to find the sum" , "Reckoning" , "Annexe" .)