The Premier League champions have had a disappointing campaign under manager Manuel Pellegrini and are down in fourth position, 12 points behind runaway leaders Chelsea with five matches left to play.
"We are not in the place where we wanted to be, we wanted to be really challenging for the title like we've done in previous years," Hart told the club website on Tuesday.
"We've got to be honest and win these last games and see where it takes us. We've let ourselves down at key moments.
"It's a difficult league to win. The two times we've won it, it's gone right to the wire," Hart added of their title triumphs in 2012 and 2014.
"This season, in a lot of games we've played well but teams have tended to raise their game against us. It's just how it is, that's not me moaning."
Hart said the defeat at third-placed United earlier this month was a particularly tough pill for him and his team mates to swallow.
"It was a difficult day, emotional to lose the derby in the manner we did after starting so well," he added, referring to the eighth-minute lead given to City by Sergio Aguero.
"There's been a lot of reflection, a lot of honest conversations. It was a tough result to take.
"This is my club. It was a bad result to lose with the great rivalry we have and the positions in the league."
City, who will finish with no silverware to show for their efforts this season, entertain FA Cup finalists Aston Villa in the league on Saturday.
(Writing by Tony Jimenez; Editing by John O'Brien)
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