Last to finish cure, getting rid of cold (6)
I believe the answer is:
endure
'last' is the definition.
('endure' can be a synonym of 'last')
'finish cure getting rid of cold' is the wordplay.
'finish' becomes 'end' ('end' can be a synonym of 'finish'**).
'getting rid of' is a deletion indicator.
'cold' becomes 'c' (eg on taps).
'cure' with 'c' removed is 'ure'.
'end'+'ure'='ENDURE'
'to' is the link.
(Other definitions for endure that I've seen before include "Bear something patiently" , "Stand" , "Suffer, tolerate" , "Undergo hardship patiently" , "'Last, bear (6)'" .)