President Hamid Karzai appealed for the two men vying to succeed him to end their stand-off over the poll result and save the country from worsening violence.
Clashes erupted in Logar province, with insurgents battling police and troops in the latest chapter of a summer "fighting season" that has seen nationwide bloodshed.
"Up to 200 Taliban attacked in Charkh district," Khalilullah Kamal, the district government chief, said. "So far, seven army and police have been wounded.
The Afghan government has been paralysed for months after the first round of the presidential election failed to produce a clear winner and the second round of voting in June triggered allegations of massive fraud.
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As fears grew of a return to civil war, the United States last month brokered an emergency deal designed to end the impasse between poll rivals Ashraf Ghani, a former World Bank economist, and former anti-Taliban fighter Abdullah Abdullah.
But neither candidate appears willing to back down, and the dispute looks set to break out again in the coming days when early results emerge from an anti-fraud audit of all eight million votes.
"I hope we stay united... So that our country is led toward peace and prosperity," Karzai said in a speech in Kabul to mark Independence Day.
"I hope that Afghanistan's election has a result soon. The people are waiting impatiently for the result.
"I hope both of our brothers... Reach an agreement so that Afghanistan soon has an inclusive government in which nobody is left out."
The election deadlock has revived ethnic divisions that lay behind the 1990s civil war in Afghanistan.
Uncertainty has also shaken the fragile economy, which is dependent on falling aid funding as the 13-year international effort to develop Afghanistan winds down.
The US has been pushing for the next president to be inaugurated before a NATO summit starting on September 4, which should sign off on follow-up support after NATO's combat mission in Afghanistan ends this year.
The audit has checked more than 50 per cent of the votes, and the next stage of invalidating fraudulent ballots will likely raise tensions between the candidates -- who are also meant to be in talks about a post-election unity government.