Is it permissible to dispose of a gift received from a Quranic competition, knowing that the reciter made mistakes in reciting the verses and no one corrected him, yet he was honored as a winner?

There is no harm in participating in Quranic competitions and receiving prizes for them. There is also no harm in utilizing the prize, because one of the objectives of such competitions is to encourage students of knowledge and elevate their level, and those organizing the competition are most knowledgeable of who deserves the prizes.