City have already missed out on UEFA Champions League and Capital One Cup honours and with Manchester United holding a 12-point advantage in the Premier League, the reigning champions look unlikely to retain their title.
Mancini is refusing to give up hope of overturning their neighbours in the title race but admits their league position has given added importance to the FA Cup clash against the Tykes, especially if he wants to deliver on his promise of silverware every season.
The City boss, who guided the club to FA Cup glory in 2011 and the Premier League title the following season, said: "It is my opinion that for a team like us it is very important to win something every year.
"It is better if it is Premier League or Champions League or Europa League but the FA Cup is important.
"These games are difficult because they don't have anything to lose and everyone can think it will be easy.
"I have watched Barnsley and it is clear they are not Real Madrid, but these games are strange.
"I don't think many people thought Arsenal could go out in a game against Bradford."