The Italian guided the Eastlands club to the 2011 FA Cup and the 2011/12 Premier League title after a barren run of silverware gains but met his demise late in the 2012/13 campaign.
As City up the search for Mancini's replacement - hotly tipped to be departing Malaga boss Manuel Pellegrini - Hart has called for massive improvement no matter how takes the reins at the Etihad Stadium.
"It's tough to talk about possibilities obviously," Hart told Goal.com.
"We had successful times under Roberto but the club's made the decision to move forward and we need to react as players.
"Whoever comes in, we'll give them everything we've got."
The England number one added: "We need to improve on next year and that's an exciting challenge.
"There's nothing massive, just little tweaks here and there. Finishing the job - we didn't win enough games - we won possession, dominated games, but didn't win them. That's what we need to improve on.
"There's a lot of work to be done behind the scenes, that's why we've got such great people in charge - strong characters, strong businessmen - hopefully they're going to make the right decisions and we're going to react as players and move forward.
"Everything's in the past now, that's the beauty of football - it's constantly changing - and we have another chance next year."