He and other senior ANC figures have been embroiled in a series of graft allegations, as South Africa has struggled with a slowing economy, high unemployment and regular violent protests.
In August, the ANC - which came to power in 1994 under Nelson Mandela after the end of apartheid - recorded its worst-ever election results at local polls.
"The ANC has heard the message that the people delivered in August. We accept that we have made mistakes," Zuma, 74, said in a speech marking the ANC's 105th anniversary.
"When leaders and members of the ANC are corrupt and steal they are betraying the values of the ANC, the people and our country. We will not allow this."
Among Zuma's possible successors are his ex-wife, African Union chief Nkosazana Dlamini-Zuma, Deputy President Cyril Ramaphosa and ANC treasurer-general Zweli Mkhize.
On Saturday, the ANC's influential women's league pledged its support for Dlamini-Zuma.
"The ANC will elect a new national leadership towards the end of the year," Zuma said.
"Too often, comrades fight for leadership positions as they see leadership as the route to material and personal gain."
As attacks on his presidency grew last year, Zuma survived an attempt by ANC rivals to oust him in November, shrugging off criticism of his conduct by the official anti-graft watchdog and the Constitutional Court.
The watchdog probe uncovered evidence of possible criminal activity in his relationship with the Guptas, a business family accused of wielding undue political influence.
Zuma, who took power in 2009, retains strong loyalty among many rank-and-file ANC party members, as well as its lawmakers.
He struck a humble note at the ANC celebrations in a sports stadium in Soweto, a hotbed of the struggle that ended white-majority rule more than 20 years ago.
"The people have told us that we are too busy fighting each other and we do not pay sufficient attention to their needs," he said.
"The ANC must unite so that we are able to unite the people against our common enemies - unemployment, poverty and inequality."
Zuma gave a shortened version of his published speech as heavy rain lashed the venue.
Copyright Agence France-Presse, 2017
Ivory Coast soldiers end mutiny
BOUAKÉ: Ivory Coast soldiers on Sunday ended a two-day mutiny in the second city Bouake and other key areas after reaching a deal on their demands for pay rises, housing and faster promotion. President Alassane Ouattara called on his compatriots to "go back about their business" as normal after the deal was accepted by the military mutineers, who actions had paralysed Bouake.
The world's top cocoa producer was rocked by two days of unrest after soldiers seized control of Bouake's streets early Friday, firing rocket-launchers and terrifying residents, in a mutiny that spread to other cities including the economic capital Abidjan. "There has been no firing since Saturday night," a local journalist in Bouake said on Sunday. "Traffic has resumed this morning and the shops have reopened."
One of the leaders of the mutiny in the west African country said he was "happy with the turn of events" after soldiers returned to their barracks.
"All we are asking is for the president to be attentive to the living conditions of soldiers," he said on condition of anonymity. In Abidjan - a bustling seaport that is home to the presidency and parliament - national television reported that shots had been fired at the Akouedo barracks on Saturday.
The soldiers had also detained Defence Minister Alain Richard Donwahi for two hours, firing Kalashnikovs and heavy weapons around the home of a senior local official where Donwahi was meeting with soldiers' representatives.
"After the regrettable events of recent days, the situation is back under control and everyone should go back about their business," Ouattara tweeted Sunday.
On Sunday, Abidjan's central Plateau area where several government offices and leading businesses are located, was calm. "Life is back to normal," a resident said. The same was true for the northern city of Korhogo and Man in the west. Ouattara announced on Saturday evening that an agreement had been reached.
In a brief televised address he said it took into account "the demands relating to bonuses and improving the living conditions of soldiers," without giving details of the accord.
Donwahi, who returned to Abidjan on Saturday night, said all bonus arrears would be paid. "We are aware that there are many problems to resolve... I can assure you that we will keep our commitments but they too should keep theirs," he said. Bouake, which is home to 1.5 million people, was the capital of a rebellion which erupted in 2002 in a failed attempt to oust then president Laurent Gbagbo.
Copyright Agence France-Presse, 2017