Rafa Benitez's side found themselves two goals down minutes into the second half after Sergio Aguero had added to Samir Nasri's opener, and were unable to produce a comeback despite Demba Ba netting a consolation.
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"We were frightened of City in the first half," he told The Sun. "We showed them too much respect, and paid the price. Maybe we might have deserved a different fate, but we started the game badly.
"We played the final 40 minutes, but that is not enough to beat City. "We had chances, and we believed we could do something, but their keeper made some good saves."
Chelsea face Fulham in the Premier League on Wednesday hoping to claw themselves back into pole position in the race for the Champions League after Arsenal leapfrogged them into third place on Saturday.
Rafa Benitez revealed that Ashley Cole will again miss out after injuring himself in the FA Cup quarter-final win over Manchester United, but the left-back is edging closer to a first-team return.
"He will not be back for Wednesday. He is getting closer but not for this game," the Spaniard said.