Period without a drink is difficult after start of drinking time (7)
I believe the answer is:
drought
'period without a drink' is the definition.
'difficult after start of drinking time' is the wordplay.
'difficult' becomes 'rough' (similar in meaning).
'after' says to put letters next to each other.
'start of' suggests taking the first letters.
'time' becomes 't' (abbreviation).
The first letter of 'drinking' is 'd'.
'rough' after 'd' is 'drough'.
'drough'+'t'='DROUGHT'
'is' is the link.
(Other definitions for drought that I've seen before include "Lack of rainfall" , "Lack of rain" , "Long water shortage" , "Period of low rainfall" , "Long dry spell" .)