Happen to live well at last, having crossed Cornish river (6)
I believe the answer is:
befall
'happen to' is the definition.
(synonyms)
'live well at last having crossed cornish river' is the wordplay.
'live' becomes 'be' (synonyms).
'at last' suggests the final letters.
'having crossed' indicates putting letters inside.
'cornish river' becomes 'Fal' (River Fal in Cornwall).
The final letter of 'well' is 'l'.
'be'+'l'='bel'
'bel' enclosing 'fal' is 'BEFALL'.
(Other definitions for befall that I've seen before include "Happen to (someone)" , "Turn out" , "Happen, occur" .)