Widely circulated video footage of Aurora officers detaining Brittney Gilliam, her six-year-old daughter, teenaged nieces and sister has sparked outrage amid a national debate over police tactics against ethnic minorities.
The police department said the officers mistakenly believed Ms Gilliam's car had been stolen and had been trained to perform a "high-risk stop".
The department has launched an investigation and will cover the cost of therapy for the children, aged six, 12, 14 and 17, it said.
On Monday, Aurora Interim Chief of Police Vanessa Wilson issued an apology and explanation for the incident, which took place on Sunday.
Ms Wilson said when officers believe a vehicle is stolen, they are trained to draw their weapons and order all occupants to exit the car and lie prone on the ground.
"But we must allow our officers to have discretion and to deviate from this process when different scenarios present themselves. I have already directed my team to look at new practices and training."
She said she has called the family to apologise and offer help - "especially for the children who may have been traumatised by yesterday's events".
"I have reached out to our victim advocates so we can offer age-appropriate therapy that the city will cover."
The family had been out to visit a nail salon, and were returning to their car after finding the salon closed.
In the footage of the incident posted by witnesses on social media, officers can be seen approaching the vehicle with guns drawn as the family is getting into the car.
Officers, surrounding the family, ordered all four girls - including Ms Gilliam's six-year-old daughter and 14-year-old niece - to lie facedown in the parking lot.
Ms Gilliam, her sister, aged 12, and another niece, aged 17, were also handcuffed.
The children can be heard crying and asking for their mother as witnesses question police about the situation. The video has been viewed over 3m times as of Tuesday.
Police have said the car's license plate had matched the number of a stolen vehicle, but from a different state. They said the misunderstanding may have also been in part due to the fact that Ms Gilliam's car had been reported stolen earlier in the year.
After realising the error, police said they released everyone, explained the situation and apologised.
Ms Gilliam told CBS Denver on Monday she does not want a police apology.
" I want change," she said. "Better protocol, better procedures because the way you did it yesterday was not it."
She added that the children were not okay. "Would your kids be okay after that? Having a gun pulled on them and laid on the ground. Especially a six year old."
The Aurora police department has also faced criticism for the August 2019 death of Elijah McClain, a 23-year-old black man who died in police custody.
Mr McClain was put in a chokehold by officers. He was eventually sedated by a medic who then noted he had no pulse. Days later, he was declared brain dead.
A coroner's autopsy found the cause of death to be undetermined. The officers were cleared of wrongdoing months later.
Mr McClain's case has seen renewed focus following George Floyd's death and the ensuing protests against racism and police violence. The state governor has appointed a special prosecutor to review the case.
In July, Aurora police banned the chokehold used on Mr McClain. New rules also say officers must intervene if they see a colleague using excessive force.