The United Nations said 2,800 people had been displaced by the clashes and many more could flee, though some were trapped.
Haftar last week ordered his self-styled Libyan National Army (LNA), which is allied to a parallel administration in the east, to march on Tripoli, the seat of the internationally recognised Government of National Accord (GNA) which is protected by an array of militias.
Witnesses said on Monday afternoon that the LNA had lost control of the old airport and withdrawn from positions on the airport road.
Eastern Libyan forces sought to reach the centre of Tripoli after their easy advance through desert hit a trickier urban phase, with deaths and displacements mounting and the West aghast at the threat to its peace plan.
The video emerged as the US temporarily withdrew some of its forces from Libya due to the "increasingly complex and unpredictable" security situation, according to Marine Corps General Thomas Waldhauser, head of US Africa Command.
U.N. spokesman Stephane Dujarric told reporters in NY on Monday that "unfortunately" the United Nations received "no positive news" in response to its urgent appeal for a truce in Tripoli.
"The upsurge in violence is further increasing the misery of refugees and migrants arbitrarily detained in detention centers in areas of active conflict", she said.
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Meanwhile, fighting was underway Monday at Tripoli's former worldwide airport, some 24 kilometers (15 miles) south of the city. The Tripoli based health ministry, on the government side, said at least 25 people including fighters and civilians had been killed and 80 wounded in fighting to the south of the capital.
Its elite Saiqa (Lightning) force, numbers some 3,500, while Mr Haftar's sons also have well-equipped troops, LNA sources say. Ahmed al-Mandhari, the World Health Organisation's regional director for the eastern Mediterranean, said in a statement Monday that the targeting of doctors was "unacceptable".
Russian Federation on April 8 called for calm on all sides.
United Nations special envoy to Libya Ghassan Salame condemned the attack on Tripoli's only functioning airport, saying the United Nations was "deeply concerned for the welfare of the civilian population in the ongoing violence".
Speaking at the EU's foreign ministers meeting in Luxembourg, Mogherini said all sides in the recent surge in fighting should "go back to the negotiating table under the auspices of the U.N".
"We need all the parties involved in this conflict to attend", Dujarric said.
Salame later Monday condemned the attack on Mitiga, saying it was "a serious violation of worldwide humanitarian law".
Fierce clashes Sunday near Tripoli saw Haftar's fighters and other powerful western Libyan armed groups exchanging fire including air strikes.