They had been hit with a two-year ban from European competitions and a €30m fine, but the Court of Arbitration for Sport (CAS) overruled the ban handed to them by UEFA.
But Carragher believes City are yet to clear their name, and reckons the suspicion around them will never go away.
“They have still lost nine games this season, which just tells you that money isn’t everything, nine teams have beaten them,” he told Sky Sports.
“Man City, with the money that they have, haven’t won the Champions League, so they’re not actually completely dominating really.
“Will it have an effect on the pitch? Will the suspicion that they have done something or have got away with something go away? I don’t think so.
“Yes, they have gone to court and won but I don’t think that suspicion will ever go away so I don’t think they have fully cleared their name in the eyes of people up and down the country.
“Whether that helps the players, they don’t have to worry that they might miss out on the Champions League.
“So they may improve and they certainly need to improve for next season.”
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