The 24-year-old playmaker joined United from Borussia Dortmund but he has yet to cement a starting place at Old Trafford.
On Saturday he made his first league start under David Moyes but the midfielder was hauled off at half-time in the 2-1 defeat to West Brom.
And Kagawa - who was the subject of interest from former club Dortmund over the summer - admits things have to get better.
"I have to take this like some kind of message (from the manager)," he said.
"I need to improve more and more, or I cannot survive at this club.
"I have to take this in a positive way and keep trying to do my best."
Kagawa feels he could have made a contribution against West Brom if he had stayed on the pitch.
"I thought that I would have been able to find some space to create the chances in the second half," he added.
"In that sense I was disappointed. Sure, I wasn't involved that much. I didn't touch the ball enough but I was trying to be patient so I didn't feel that bad."