Burnley captain Ben Mee admitted he was ashamed after a banner was flown over the Etihad Stadium during the team's 5-0 thrashing by Manchester City on Monday evening.
A plane carrying the message 'White Lives Matter Burnley' flew over the ground in the early stages of the team's first Premier League outing since the competition's restart.
It occurred just moments after all players had taken the knee in support of the Black Lives Matter movement at kick-off.
The club issued a strongly worded statement before the end of the game and Mee revealed his embarrassment afterwards and felt it had affected the team's performance.
"We as a group of players condemn it, we're ashamed, we're embarrassed," Mee told BBC Radio 5 Live.
"It definitely had a massive impact on us to see that in the sky. We were embarrassed, disappointed, upset. We are embarrassed that our name was in it.
"That they tried to attach it to our club. It doesn't belong anywhere near our club. Fans like that don't deserve to be around football."
Clarets manager Sean Dyche, who did not see the plane at the time, was also disappointed about the incident.
Dyche said: "As a club we can only apologise. It is unacceptable."
The club statement added the perpetrators were not welcome at Turf Moor and would be banned for life if identified.
The statement released at half-time read: "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.
"The club has a proud record of working with all genders, religions and faiths through its award-winning Community scheme, and stands against racism of any kind.
"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."
Former Burnley skipper Frank Sinclair said on Twitter he was disappointed by the incident.
"Please understand the message #BlackLivesMatters is for the good of humanity not just Black lives," he said.
Anti-discrimination group Kick It Out released a statement in support of the BLM movement, saying the purpose of it was not to "diminish the importance of other people's lives".
The statement read: "Fighting for equality requires addressing injustice - imbalances that plague our society and prevent everyone from receiving fair treatment.
"The point of Black Lives Matter is not to diminish the importance of other people's lives. It is to address an imbalance and highlight that black people are being denied certain human rights simply by virtue of the colour of their skin.
"It is about equality. We shall continue to support the Black Lives Matter movement and the fight for greater equality in football."
The Burnley chairman has vowed to "root out the racists" behind the banner.
Clarets chairman Mike Garlick told ESPN: "We will root out these racists and I am sure they will be dealt with in due course.
"We will do whatever we can to make sure justice is done against these people.
"Let's be clear. The plane didn't fly in our name. It has absolutely nothing to do with Burnley Football Club and we absolutely deplore it.
"But we have to talk about it. We cannot and will not sweep it under the carpet.
"I was at the game and I have been chairman/co-chairman for eight years. I have always been proud to be in that position, but I can tell you that I wasn't proud to be Burnley chairman last night."
Lancashire Police said it was looking into the incident to see whether any criminal offences had taken place.
The aircraft took off and landed at Blackpool Airport, which said it would stop operating banner flights in the wake of the incident.