Starts to eat -- needs drink, so finishes (4)
I believe the answer is:
ends
'finishes' is the definition.
('end' can be a synonym of 'finish')
'starts to eat needs drink so' is the wordplay.
'starts to' indicates taking the first letters.
The first letters of 'eat needs drink so' is 'ENDS'.
(Other definitions for ends that I've seen before include "deaths" , "Extremities" , "They may justify the means" , "limits" , "Brings to a close" .)