Burnley captain Ben Mee says he and the squad were left "ashamed and embarrassed" by a far-right message flown above the Etihad.
A plane was flown above the stadium pulling a banner with the racist phrase 'white lives matter'.
Burnley issued an apology for the incident mid-game, promising lifetime bans for those responsible.
Players and match officials across the Premier League have taken a knee at kick-off in each game, while players' names have been replaced on jerseys with the 'Black Lives Matter' phrase.
Mee was interviewed by Sky Sports after Burnley's 5-0 defeat to Manchester City, and took it upon himself to first address the racist incident.
"I'm ashamed and embarrassed that a small number of our hands have decided to fly that around the stadium," the defender said.
"They have completed the point. There's a group of lads in there that is embarrassed to see that, they've missed the point of the whole thing we are trying to achieve and these people need to come into the 21st century and educate themselves.
— Sky Sports Premier League (@SkySportsPL) June 22, 2020
"It does not represent what the club is about, what the players are about and what the a majority of the fans are about, that's for sure. It's a small minority of people and I'm really upset that happened."
Mee told Sky that the club were aware such a racist incident may occur, "Literally as we were coming out, we heard some whispers that it might happen, the club were trying to stop it.
"I hope it doesn't happen again. We are ashamed and upset."
Burnley Football Club strongly condemns the actions of those responsible for the aircraft and offensive banner that flew over The Etihad Stadium on Monday evening.
We wish to make it clear that those responsible are not welcome at Turf Moor.
This, in no way, represents what Burnley Football Club stands for and we will work fully with the authorities to identify those responsible and issue lifetime bans.
We are fully behind the Premier League’s Black Lives Matter initiative and, in line with all other Premier League games undertaken since Project Restart, our players and football staff willingly took the knee at kick-off at Manchester City.
We apologise unreservedly to the Premier League, to Manchester City and to all those helping to promote Black Lives Matter.