Encouraging a number to take part in audition (10)

I believe the answer is:

heartening

'encouraging' is the definition.
(heartening is a kind of encouraging)

'a number to take part in audition' is the wordplay.
'a number' becomes 'ten' (ten is a kind of number).
'to take part in' means one lot of letters goes inside another.
'audition' becomes 'hearing' ('hearing' can be a synonym of 'audition').
'ten' put within 'hearing' is 'HEARTENING'.

(Another definition for heartening that I've seen is " Cheering".)

I've seen this clue in The Telegraph.

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