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Elliott Bennett and Javier Garrido each came close with early efforts as Norwich started brightly against the bottom-of-the-table Royals.
Alex McCarthy pulled off a string of fine saves shortly after the interval, but Ryan Bennett pounced on his parry to drill the ball into the roof of the net from close range on 50 minutes.
His namesake, Elliott, soon doubled Norwich's advantage when he calmly converted a rebound after McCarthy could only palm a Martin shot into his path.
McCleary was Reading's standout performer all afternoon and added a goal to his impressive performance when he carried the ball up to the edge of the area before shooting past Bunn.
It was not enough for Reading to collect any points though, and they are now 10 points away from Premier League safety with four games remaining.