The Japan international returned to the starting lineup in Wednesday's Capital One Cup win over Liverpool, but his manager believes there is more to come from the 24-year-old.
The Japan international has been on the periphery so far this season under Moyes, having failed to make the squad for the Red Devils' trip to Anfield earlier this month and featuring only as an unused substitute in the 4-1 defeat against Manchester City last weekend.
Kagawa was restored to the starting lineup for the 1-0 victory over Brendan Rodgers’ side, and even though he displayed glimpses of the talent that persuaded Sir Alex Ferguson to pay Borussia Dortmund 20 million euros for his services following the 2011-12 season, Moyes still believes the 24-year-old Kagawa has significant room for improvement.
"He did OK. I thought he grew into the game a little bit more," Moyes told reporters after the victory which set up a fourth round clash at home to Norwich. "There were bits which I was pleased with, and bits where we were looking for a little bit more. Overall, he's still getting back.
"A lot's been said [about his lack of playing time], but I think most people who watched would say he's still short of being really at full match fitness."