Clattenburg sent off Everton duo Tony Hibbert and Phil Neville for conceding second-half penalties as the Reds won 2-1 at Goodison Park.
But Moyes was incensed after his team were denied a late penalty when Jamie Carragher appeared to foul Joleon Lescott in the box, while Kuyt was only booked for a two-footed lunge at Neville.
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