Moyes continues to find his new position at Old Trafford scrutinised with the club seventh in the Premier League, out of both domestic cups and with a two-goal deficit to overcome if they are to stay in the Champions League.
Fans have witnessed five defeats at Old Trafford since the former Everton manager took over from retiring Sir Alex Ferguson in the summer but Moyes has promised improvement.
Moyes's letter in the club's 12th Man magazine read: "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 - and my players, staff and I are desperate to compensate for that.
"You are accustomed to seeing a successful Manchester United and the backing you have given the players and me throughout the season has been incredible.
"Away from home the travelling fans have remained the best in the country while at Old Trafford your unwavering faith has been noticeable and hugely welcomed.
"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.
"Everywhere we turn people outside the club have a lot to say about Manchester United, but we have all stuck together through some tough times this season and I firmly believe that, in the long run, we will all come out at the other end, stronger for the experiences.
"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.
"A big step towards that will be transforming Old Trafford back into the fortress it has long been renowned as, and we will do everything in our power to get the positive performances and results to do that, so that we can give you season ticket holders and all of our supporters something to shout about."
Moyes takes his players to West Brom on Saturday afternoon, where a victory would lift them above Everton - otherwise engaged in the FA Cup quarter-finals - into sixth.