Is it permissible to listen to the Qunut supplication from a mobile phone during the night vigil (Qiyam al-Layl) instead of reciting it, knowing that it achieves devotion?

Reciting Qunut in Witr prayer is a sunnah (definition) (recommended), not an obligation. It is sunnah to do it sometimes and omit it sometimes, and there is no sin upon one who omits it. If you have not memorized it, it is permissible to read it from a phone. Listening to it is neither required nor does it invalidate the prayer.