City boss Roberto Mancini had hoped United would progress at the expense of Real Madrid this week to provide a distraction from their increasingly strong Barclays Premier League title march.
City need to overturn a 12-point deficit in just 10 games to retain their title and with United now concentrating fully on domestic matters, the FA Cup seems their best hope of silverware.
Mancini said: "It is clear now it is worse for us because they can put all their strength in the championship.
"It was better they stayed in the Champions League. When you play an important game every three days it is more difficult.
"But it was difficult (for us) before the game against Madrid.
"I think they have an advantage because 12 points are too much but our job is to try until the end of the season."
Mancini raised expectations at City when he guided them to FA Cup glory in 2011, ending a 35-year trophy drought.
The Italian, who has fielded strong sides in the previous rounds against Watford, Stoke and Leeds, is intent on a repeat victory.
He 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."
The other way of looking at United's European exit - and a potential FA Cup defeat to Chelsea on Sunday - is that it could knock their season off the rails.
Mancini concedes he cannot see that happening.
He said: "We hope but we don't know this. Sometimes it is possible but they are strong enough."
Barnsley travel to the Etihad Stadium for the Saturday evening clash as rank outsiders - some bookmakers quote odds as long as 18/1 on a Tykes victory - due to their position in the Championship relegation zone.
Although Mancini will not be risking key players Vincent Kompany and Sergio Aguero due to injury, he will not be taking the game lightly.
He said: "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."
Mancini was in good spirits at his weekly press conference, feigning surprise the first question was not about Mario Balotelli and was then bemused to discover it was actually about International Women's Day.
But on the serious business he stressed the importance of remaining strong throughout the remainder of the campaign and reiterated his determination to fight United until the end.
He said: "This is our job. I think it is important to be strong in a difficult moment.
"You should try and win every game and take the championship.
"I used to play until the last game. I won and lost a championship with seven, eight, nine, 10 points more.
"In football, two or three games can change everything. United have won 14 of the last 16 games but if they have two or three draws in a row, it can change."
Mancini did not dismiss speculation striker Edin Dzeko could leave the club in the summer but he insisted, for now, the Bosnian remains part of his plans.
The 26-year-old, signed for £27million from Wolfsburg two years ago, has been linked with a return to German football with Borussia Dortmund.
Mancini said: "At the moment, there is nothing. In the end, maybe. I don't know.
"In the summer it could be possible but for Edin, for other players, we don't have anything.
"Dzeko is an important player for us. We want that he stays here."
Kompany remains on the sidelines after six games out with calf problems while Aguero pulled out of Monday's win at Aston Villa with a knee injury.
Mancini hopes both will be fit to face Everton next week but midfielder Jack Rodwell, who limped off at Villa Park with yet another hamstring injury, could be out for a month.
The 21-year-old, whose City career since a £12million summer move from the Toffees is yet to take off, has a history of such problems and only returned to action in January after a three-month lay-off.
Mancini said: "Jack is so-so. I think we need two or three weeks after the international break."