The playmaker conceded he was below-par in Japan's two friendly losses to Serbia and Belarus this week as he is struggling for form and fitness.
Kagawa has started just three games for United this term under Moyes with the 24-year-old also being taken off at half-time of the recent defeat at home to West Brom.
The former Borussia Dortmund star believes he needs to be playing more if he is to recapture his best form for club and country.
"There's a matter of fitness and there's also that atmosphere of playing for my country," Kagawa said. "It's different from the atmosphere at United, but you still get a good feel for being on the pitch when you play regularly.
"Maybe it's because I'm not used to being on the pitch of late, but I didn't feel like I was able to enter these two games properly.
"During tough times like these I have to improve myself.
"Personally, I felt strongly after these two matches (against Serbia and Belarus) that I need to play more and, as a team, there are many things we need to do.
"We all need to go back to our European or J-League clubs and take what we learned in these two matches and work to fix those problems."