Manuel Pellegrini's side lifted silverware by beating Sunderland at Wembley earlier this month, but Clichy believes that City must edge out their rivals in the race for the Premier League title to be happy with their campaign.
"Just the Capital One Cup won't be enough," Clichy told reporters. "If you don't have this energy and desire for winning every single game no matter who you play, then you have nothing to do in City.
"So the project is fantastic, we are lucky to be here sharing this with the team-mates but, you know, its crunch time and we have to get the points until the end of the season."
City are currently fourth in the Premier League, six points behind leaders Chelsea with three games in hand.