Manchester City manager
Richards suffered the setback when on Olympic Games duty for
City initially thought Richards would only be out of action for a week but the prognosis is worse than first feared.
Speaking after the 3-2 win over Chelsea at
"He has a problem with his ankle, a big problem maybe, and it may be one month out."
Mancini said: "Joe had a problem yesterday with his back but I don't think it is serious."
Mancini believes City will see the best of striker Carlos Tevez this season because he has had a full pre-season under his belt.
Tevez scored a stunning goal as City came from behind to win 3-2 against Roberto Di Matteo's side.
Mancini said: "I am very happy because Carlos this year has worked very well in pre-season, and his form is better than last year.
"Also this year he wants to play football. For a player, it is difficult when he does not play. Last season he did not play for six months and this is difficult.
"I expect him to be a very important player. He is a top player if he wants to play.
"Last year we had Carlos for the last 10 games and it was important because he scored five or six goals.
"But he was not 100 per cent. This year is the first time in four or five years he has done a pre-season and, for every player, this is important."
Yaya Toure and Samir Nasri were City's other scorers with Fernando Torres and Ryan Bertrand netting for Chelsea.
But Mancini accepted the dismissal of Chelsea defender Branislav Ivanovic late in the first half for a challenge on Aleksandar Kolarov was a key moment in the game.
He said: "It was a good performance. When Chelsea were leading 1-0, we played better.
"With the sending-off, it was difficult to see from the bench but it is an advantage to play with an extra player.
"It had an impact on the game. It is impossible to say no because 11 versus 11 is different. But we played better than Chelsea, played better football."
Chelsea boss Roberto Di Matteo claimed the red card issued by referee Kevin Friend was the key moment in the game.
He said: "It changed the game in favour of Manchester City. We competed very well and at that point were in the lead.
"It was difficult in the second half against 10 men.
"The challenge had no intent to hurt the opposition. It was a sliding tackle and he got the ball as well.
"Sometimes a red card is given in those circumstances, sometimes it is not. But there was no intention to harm the opponent."
Di Matteo was happy with the performance of new signing Eden Hazard but believes all the players have a responsibility to fill the gap left goals-wise after the departure of Didier Drogba.
He said: "He was lively, caused a few problems, had a couple of opportunities, we saw glimpses of his quality, and he will be a big success for us.
"Fernando (Torres) was also dangerous in what was his first 90 minutes since coming back into training under three weeks ago.
"But we have to share the responsibilities for goals between the team - Hazard, Torres, Ramires, (Juan) Mata, (Daniel) Sturridge, (Frank) Lampard can all score goals."