A Russian-born man opened fire on passers-by with an automatic rifle on Saturday before barricading himself in a supermarket with hostages, where he was shot dead by police.
Police stormed the supermarket after unsuccessfully trying to negotiate with the suspect for 40 minutes, although officers were initially filmed running away from the incident, prompting the resignation of a police chief.
Klitschko said the wounded child, whose parents were killed in the shooting, was in moderate condition while one of the adults was critical.
"They are all receiving all necessary medical care," the mayor said on Telegram.
President Volodymyr Zelenskiy said on Saturday that the shooting, which happened in the capital's leafy Holosiivskyi district, injured 14 people.
The supermarket has been cordoned off and remains closed.
Bullet holes are visible in windows of the supermarket, and bloodstains can be seen nearby.
Flowers were left near a residential building a couple of hundred metres from the supermarket, where the shooter shot his first victims.
"I saw how people grabbed children from the playground and ran away. They screamed: 'run away, hide'," Daryna, a 31-year-old local resident, told Reuters.
"People didn't understand what was going on. They said that there was a man there, a man was shooting with a machine gun."
Shootings of this nature are extremely rare in Ukraine and the country's security service said the incident was being investigated as a terrorist act.
Police have not yet identified a motive for the crime.
Yevhen Zhukov, the head of Ukraine's Patrol Police - a division of the country's police service whose duty is to patrol the streets - resigned on Sunday after social media circulated a video showing patrol officers running away after hearing gunfire, leaving civilians without protection.
Reuters could not independently verify the video.
"The police officers acted unprofessionally and disgracefully. As police officers, they should have been helping and rescuing our citizens. But they failed to assess the situation properly and left civilians in danger," online media outlet RBC Ukraine quoted Zhukov as saying.
"As a combat officer, I have decided to submit my resignation from the position I currently hold," Zhukov added.
Earlier on Sunday, minister Klymenko said the officers' behaviour was "a disgrace to the entire system".
An investigation has been launched and decisions will be made regarding their superiors.
The shooting of civilians not far from central Kyiv has raised questions both about the public's right to self-defence and about how the gunman was able to obtain a firearms permit, Interior Minister Ihor Klymenko said.
"The attacker's mental state was clearly unstable. How he obtained the medical certificates required to renew his gun permit must be thoroughly investigated," Klymenko said.
He said the ministry intends to prepare the final version of a bill on civilian firearms as he was certain that "people should have the right to armed self-defence."
At present, Ukrainians can own only hunting weapons.
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