Demonstrators held hands as they marched through Harare in what organisers described as a peaceful protest.
However, riot police used batons and chased demonstrators away, while also lashing out at passers-by, a reporter for the AFP news agency said.
Lovemore Madhuku, the NCA leader, was arrested early on Tuesday before the protest took place. The group says he has not yet been released.
"The NCA emphatically condemns this unjustified obstruction of the organisation's peaceful protest actions," the group said.
The World Food Programme has warned that it will have to cut food rations in Zimbabwe, where more than five million people are expected to require food aid by January.
Donors have failed to provide the funds needed provide the aid, the agency said.
With inflation running at more than 231 million per cent, half of the population requires emergency food aid while a breakdown in basic services has led to outbreaks of cholera in Harare.
Political violence 'soars'
The latest clashes between civilians and police came as Zimbabwean human rights lawyers said state-sponsored political violence was on the rise in the southeast African nation.
Zimbabwean Lawyers for Human Rights said its members recorded 1,300 incidents of political violence in September this year - an increase of 39 per cent on the previous month.
Violent attacks catalogued by the lawyers' group and other organisations ranged from rape to property disputes to murder.
The human rights groups, who believe the police are responsible for most of the violence, are backing Tsvangirai's stand against Mugabe's attempt to retain control over the interior ministry - which remains a key stumbling block to forming a government.
Southern African leaders on Sunday suggested that Mugabe should retain partial control over the ministry, which oversees the police, in order to push ahead with the power-sharing government.
But Tsvangirai rejected the proposal, saying Mugabe's "utter contempt" for his party would render the deal unworkable. He accused the Southern African Development Community membership of lacking the "courage and decency" to force Mugabe to accept a fair deal.
Speaking on state television, Mugabe announced on Tuesday that he intends to form a new government "as soon as possible".
A Zanu-PF minister said Tsvangirai would be invited to nominate members to a joint cabinet.