Praising the "magnificent" fans for their patience, the Scot, who was hand-picked to succeed Sir Alex Ferguson last summer by the man himself, admitted that their struggles have taken him by surprise but is adamant that things will get better.
"While I knew that this job would be a challenge when I took it on, the difficult season we have experienced was not something that I envisaged, which I am sure is the case as well for you supporters," wrote Moyes.
"My players, staff and I are desperate to compensate for that.
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"Supporting your team when they are winning is easy but much harder when things are not going as well, and the loyalty you have shown us has been magnificent."
The Red Devils, who won the Premier League last season, sit seventh after 27 games. They are 12 points behind fourth-placed Manchester City, having played a game more, and stand on the brink of an exit from the Champions League after losing the first leg of their knockout tie with Olympiakos 2-0.
But Moyes vowed: "Everything we have been through will make us a better, stronger team and club in the future.
"Over the years you have seen great winning sides here and, in time, I have absolutely no doubt that we will see great winning sides here again."