Without hesitation, asks if you'd care for a drink (4,1,4)
I believe the answer is:
like a shot
'without hesitation' is the definition.
(I've seen this in another clue)
'care for a drink' is the wordplay.
'care for' becomes 'like' (I've seen this in another clue).
'drink' becomes 'shot' (I've seen this before).
'like'+'a'+'shot'='LIKE A SHOT'
'asks if you'd' is the link.
This may not be correct. Some or all of it may be part of another bit of the clue.
Can you help me to learn more?
(Other definitions for like a shot that I've seen before include "right now?" , "enthusiastically?" , "Eagerly" , "looking round, in field event?" , "'Pronto!'" .)