Thomas Cashman shot the youngster as he chased a drug dealer who tried to run into her home in August 2022.
He was sentenced to life with a minimum term of 42 years in April 2023.
Yesterday three Court of Appeal judges threw out his bid to challenge his conviction.
It came a year after the same court dismissed an attempt to appeal his sentence.
Barristers for Cashman, 36, had asked the judges to look into claims that jurors were given panic alarms by police, as well as allegations that evidence was given to them during their deliberations that was not presented at trial.
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