Encourage to listen to half a score (7)
I believe the answer is:
hearten
'encourage' is the definition.
(I know that hearten is a more specific form of the action encourage)
'listen to half a score' is the wordplay.
'listen' becomes 'hear' (I've seen this before).
'to half a score' becomes 'ten' (I've seen this in another clue).
'hear'+'ten'='HEARTEN'
'to' acts as a link.
(Other definitions for hearten that I've seen before include "Inspirit" , "Encourage or cheer in cardiac manner" , "Comfort" , "Be encouraged" , "Make (someone) more confident" .)