Third-bottom Sunderland were thrashed 4-1 by Arsenal at the Emirates Stadium and Poyet said they could be left red-faced if they played that badly against City at Wembley next Sunday.
"Wembley could have been on the players' minds but this was a very good wake up call. If we play like this against Manchester City we could be embarrassed," Poyet told the BBC.
Sunderland, who have performed far better in the cups this season than in the league, fell behind after five minutes against Arsenal when Olivier Giroud finished off a neat passing move.
The French striker added another thanks to some woeful defending before Tomas Rosicky scored and Laurent Koscielny was left unmarked to head home a fourth in the second half.
"The combination of Arsenal playing well and us being poor on the ball made it difficult for us," added Poyet, whose side beat Southampton to reach the FA Cup quarter-finals last week.
"We needed to be organised and see the danger but they punished us. Then we made a mistake for the second goal, which we never make, and that made it tough.
"Then we had to make sure we didn't concede another three or four in the second half because that would have been embarrassing.
"Sometimes you need to accept that you are second best. Credit to Arsenal."
Manchester City, who beat Stoke City 1-0 on Saturday to stay third in the Premier League, have scored 118 goals in all competitions this season.
(Reporting by Toby Davis; editing by Ken Ferris)
