Other 20 police officers sustained injuries.
The officer died in Kisumu while the civilian was shot dead by police in the Embakasi area.
Four other civilians are nursing wounds in the hospital after police shot them.
Property of unknown value was destroyed as police clashed with the protesters.
The marches had been peaceful until when police intervened and tried to repulse the group in the area.
Police said that Ben Oduor who was attached to Keroka police station in Nyamira County was seriously injured while on duty in Kisumu Azimio demonstrations at the Jamia supermarket in Kisumu.
But other witnesses said he was run over by a police car as he tried to escape from an irate mob.
This was after he missed a step as he tried to board their vehicle.
Inspector General of police Japhet Koome said police were overwhelmed by the rioters before he was injured.
He was administered at Aga Khan ICU where he died at around 10:15 pm Thursday.
The National Police Service (NPS) has condemned the incident saying investigations have already been launched.
“As a result of yesterday’s demonstrations, one police officer in Kisumu succumbed to injuries sustained while on duty, 20 officers were seriously injured as they repulsed violent demonstrators,” Koome said in a statement.
Koome lauded the police for their commitment, resilience, and acting in the law in the efforts to quell the chaos yesterday.
However, human rights groups have maintained that the officers are applying excessive force to the demonstrators as scores were injured while a number were reported dead.
The IG said they would not be intimidated by threats and they will serve without fear or favor.
