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Manuel Pellegrini's side crashed to a 2-1 loss in front of their own fans after going down to nine men, leaving them bottom of Group E with two games remaining against Bayern Munich and Roma, respectively.
Having let a 2-0 lead slip away at the Russian side a fortnight ago to draw 2-2, Clichy concedes that the team have found Europe a tougher proposition than the Premier League, and admits that they can't just rely on the likes of Sergio Aguero, Yaya Toure and David Silva to produce moments of magic.
"It’s difficult because in England, we are one of the best two or three teams,” he told Sky Sports News HQ. “In the UEFA Champions League, it is different because teams play in a different way.
"People may assume we are better than CSKA, but as the manager said, they are one of the best teams in their country. They know how to play in this competition and if you allow them to have time on the ball, they can hurt you.
“In the Premier League, we don’t always find that is the case. It is not that we are above everyone, but sometimes we rely on Sergio [Aguero] doing something amazing or Yaya [Toure] or David Silva producing a piece of magic because we have so much quality within our team.
“In Europe, you don’t have that luxury because if you rely on one of your own players producing something special, it could be too late because the opponents will almost certainly be capable of hurting you, so that is the balance we need to find.
“European football is a different ball game. Europe is less intense, less physical than the Premier League and is more technical and we need to adapt to that.”
CSKA midfielder Pontus Wernbloom, meanwhile, has revealed the secret to his team's vital win at the Etihad Stadium, rubbing salt in the wounds by suggesting that City are too predictable.
"We knew that they were going to come against us with big numbers and we know that they are vulnerable to counter-attacks," he told UEFA's official website.
"We've played them four times now in two years and we know they're going to give us space because, like I said before, they come forward in big numbers - almost eight men who are staying with two central defenders.
"We knew that Doumbia is faster than them and if we just got the first pass away from them when they pressed us, we could counter them. We managed to do that, especially in the first half."
Meanwhile, City loanee John Guidetti has promised Celtic fans that there is more to come from him after he picked up the SPFL Player of the Month award for September.
The Swedish striker hit two goals in three Scottish Premiership matches last month, as well as scoring a hat-trick in a 6-0 League Cup win over Partick Thistle, and is now averaging a goal a game for the club.
His fine form rewarded with a call-up to the senior national team this week, manager Ronny Deila has revealed that he would be interested in signing his star man permanently when his City contract expires at the end of the season, but the 22-year-old isn't looking that far into the future at this point.
"I just try to do my best and it’s going well," he told Sky Sports. "I’m playing with great players in front of amazing fans who help me, and the manager believes in me, and has given me this opportunity.
"It didn’t go too well for me for two years because I was injured - it wasn’t because I was playing badly. It made me stronger as a person & probably made me a better footballer. It makes me so much more grateful for the position I'm in now.
"I’m playing with one of the biggest clubs in the world. It’s just an absolute pleasure and it’s an honour to put on the Celtic jersey. I just want to cherish the moment I’m in and just enjoy every day and have a smile on my face, go in and play for Celtic.
"I hope the fans have still to see the best of me. There are big games coming up which are important and I want to do well in them."