City, beaten 3-2 by Real Madrid at the Bernabeu in their Group D opener on September 18, are looking to get their European season kick-started on Wednesday when they take on German champions Borussia Dortmund at the Etihad Stadium.
"This is the second year for us in the Champions League and the second year that we got the worst group," Mancini said. "Like we have improved (over the years) in the Premier League, we need time to improve in the Champions League, but we don't have a lot of time - only five more games, and we need to improve quickly."
With Dortmund having begun with a 1-0 win over Ajax, the pressure is on the Blues to secure all three points if they are to avoid a repeat of their maiden Champions League campaign last term, which ended in failure to qualify from the group stages.
The Italian is optimistic, though, that if City do manage to up their game and make it beyond the group stage, there is no reason why they cannot then go all the way.
"We need to not concede easy goals to drop points like we did against Real Madrid. If we want to go through, we need to improve very quickly," he added.
City midfielder Jack Rodwell is fit for Wednesday's contest after missing two games with a groin problem, but full-backs Maicon (heel) and Micah Richards (ankle) are not yet ready to return to action, while centre-back Kolo Toure has not been registered for the competition.
It remains to be seen whether Joleon Lescott will be selected after he was dropped for the Real clash and once again did not feature in Saturday's 2-1 league win at Fulham. The England centre-half played a key role in City's title success last season, but is now in a more uncertain situation.
Asked if he had lost faith in Lescott, Mancini said: "He is one of the best defenders. But I have six or seven defenders and I made a different choice last week and in Madrid.
"Tomorrow I will decide again, but Joleon is an important player for us."
Source: PA
Source: PA