Manchester United head into Tuesday's Champions League meeting with Roma at Old Trafford having gone nine matches without a win, leaving Carrick under increasing pressure and the club just above the Premier League relegation zone.
The 3-0 home defeat against Bournemouth prompted further frustration among supporters, but Mainoo insists the squad must take responsibility and repay the faith shown by their head coach.
"I feel like our way to repay that is to also be behind the manager on the pitch and off it, and fight for us as a team," the midfielder said.
"In these tough times you can't expect the fans 100,000 per cent of the time to be happy and cheering when we're not putting out results, you know. We owe it to them to put the results out there for them to cheer again."
Mainoo believes a positive result could quickly change the atmosphere around the club, despite the difficult run that has left United searching for confidence.
"Of course, it's been a tough season. We've not had the results we've wanted, but you know we need to look forward now. There are big, big games ahead and I feel like we're one win away from changing that."
The 21-year-old has experienced both the highs and frustrations of life at United, having come through the academy and established himself in the first team. He signed a new five-year contract in April after regaining his place under Carrick, having previously asked to leave during Ruben Amorim's tenure.
His relationship with the current manager has been particularly important, with Mainoo grateful for the trust and guidance he has received since Carrick's arrival.
"I'm very grateful to the boss for putting the trust in me as soon as he came in, putting that belief in me and giving me all the advice that he has," he added.
"Knowing he's a player who's been at this club and knows the history of this club and knows what it takes to win, I take those words to heart and I try and apply them every day in training and in games."
Mainoo also feels a responsibility to represent the values of the club, having progressed through its youth system and witnessed what Manchester United means to supporters.
"I feel like I take some responsibility in that and in spreading the word of what is Manchester United, coming up through the academy. I plan to send that message for hopefully as long as I can."
Having returned against Chelsea after missing five matches, Mainoo is now looking to contribute more consistently as United attempt to arrest their slide.
"I've tried to take on more responsibility in the team. I want to impose myself more in games and they've definitely helped to do that, giving me more responsibility with taking the ball."