Arab-owned City jetted out to Abu Dhabi following last Sunday's 2-0 defeat by Arsenal for a week's warm-weather training before arriving back in Manchester at around 2000 GMT on Friday.
After a goalless first half at the Etihad Stadium, City midfielder Fernando's defensive error allowed on-loan Chelsea striker Patrick Bamford to put Middlesbrough in front on 53 minutes.
City continued to be frustrated by the Championship (second tier) side and with time running out Middlesbrough substitute Kike added a second to seal the victory.
Despite the embarrassment of being knocked out by lower league opposition, City boss Pellegrini refused to blame his side's late arrival from Abu Dhabi as the reason behind their poor performance.
"No, I don't think so," he told the BBC. "It's easy to have an excuse, but I don't think that was the case.
"Before they scored the first goal, we had five or six clear chances. They didn't have one chance. But we conceded a very easy goal, and the game changed. We were nervous.
"They are a team who work well; that's why they are second in the Championship. They had opportunities to score more goals and deserved to go through."
Premier League champions City, who trail leaders Chelsea by five points at the top of the table, travel to Stamford Bridge next Saturday in a mouth-watering clash against Jose Mourinho's side.
With the Premier League and Champions League the only silverware left for faltering City this season, Pellegrini urged his side to put their FA Cup disappointment behind them amid accusations they took the Middlesbrough game too lightly.
"I wanted to continue in all the competitions but now we must focus on the Premier League and Champions League," the Chilean said.
"We now have a difficult game against Chelsea, we must reduce the five-point gap."
(Reporting by Michael Hann)
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