The Argentine scored three goals in last weekend's 5-0 FA Cup quarter-final victory over Barnsley two days after being arrested for driving while disqualified and the Premier League champions could do with more of his goals at Goodison Park on Saturday (1245 GMT).
They face a team against whom they have a poor recent record at a time when they need to start making inroads on the 12-point gap on leaders Manchester United if the second-placed side are to keep alive faint hopes of retaining their title.
Despite the serious task ahead, Mancini managed a light-hearted mood at his news conference which he started by donning a red nose in support of British charity Comic Relief.
Having determined it was rather tricky to speak wearing the item, he removed it but continued to raise smiles.
"The police didn't stop Tevez yesterday?" he asked. "It's a problem for tomorrow. If they (had) stopped Tevez maybe tomorrow Tevez can score three goals."
Seventh-placed Everton are a difficult prospect for City with David Moyes' side having won five of their last seven meetings and drawn 1-1 at the Etihad Stadium in December.
"Every game against Everton is very hard," Mancini said. "Everton is a top team, they play always 200 percent. They are strong, they have good players, a good manager."
Mancini is still without captain Vincent Kompany and forward Sergio Aguero but both should return from their injuries for the league match against Newcastle United on March 30.
Kompany has been absent with a calf injury since the end of January and Aguero suffered a knee injury earlier this month.
"Sergio is in Argentina but he comes back on Sunday," the Italian said. "Vincent is not in our plans for tomorrow. But I think Vinny will be ready, like Sergio, for the Newcastle game."
Mancini hopes to have narrowed the gap on neighbours United by the time of the Manchester derby on April 8.
"The next two or three games will be crucial," he said. "We hope we can play the derby with the lead down to 10 or nine or seven points," said Mancini, whose team have 59 points from 28 games with 10 matches remaining.
"When you are behind you need to win every game. You need to do everything."
City defender Pablo Zabaleta said his side, who have conceded only 24 goals compared to more than 30 for every other team in the league, would have to be at their defensive best against Everton and in-form Marouane Fellaini.
"Fellaini is very good and a very tall player so he can be difficult to deal with," the Argentine said on the club's website (www.mcfc.co.uk).
"They play a lot of long balls and are very strong in set pieces. Their full backs overlap a lot, especially Leighton Baines on the left side and he can deliver the ball very well.
"So, defensively we have to be very organised and fight for every ball and with the quality we have we need to create and take our chances when they come along."