The Scot has endured a difficult first season at Old Trafford since being charged with the task of replacing the legendary Sir Alex Ferguson and this week wrote to season-ticket holders to thank them for their patience and support.
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The Old Trafford outfit's many struggles and setbacks have called the players' commitment to Moyes into question, with suggestions that he does not have the squad's backing, but the United boss rejected that notion.
"I don't know if it's the right thing [to state] that they're not playing for me. I think they definitely do, so I don't accept that," the former Everton boss said.
"I've been honest with the players and the media, I've tried to be straight as I can. I've done it with the supporters.
"We need to play better. I think Olympiakos was the poorest we've played. That was disappointing but we will improve from that and we will be better.
"We had a bad night and we all take responsibility for that. But I refute that was the worst night or all those sort of things.
"We didn't play well, we said that honestly and we know we have to do much better. The players are aware of that and we will have to do better in the coming months."