Cornish listener's promise? (5)
I believe the answer is:
swear
'promise?' is the definition.
(as in swearing something is true)
'cornish listener's' is the wordplay.
'cornish' becomes 'SW' (Cornwall is in the South West of England).
'listener' becomes 'ear' (body part which hears or listens).
'sw'+'ear'='SWEAR'
(Other definitions for swear that I've seen before include "Vow, attest" , "Use bad language" , "Curse" , "Assert" , "Express annoyance" .)