Villagers were streaming out of the violent zones adding to the tens of thousands already displaced in the area, said Patrick Lavand'homme, a UN humanitarian official.
The army launched counter attacks on Monday after Nkunda's fighters claimed to have captured an area around mobile phone company installations that cover the area.
"We restarted the offensive early this morning," said colonel Jonas Padiri, a military spokesperson.
Fighting in the past two days has killed four soldiers and injured about 20, according to officials.
There were no casualty numbers given for Nkunda's soldiers.
Mushake is situated on a hill about 10km west of Sake and has been held by Nkunda's troops since August.
Major Vivek Goyal, a military spokesman for the 18,000 strong UN peacekeeping mission in DR Congo, said fighters had put up "strong resistance" and had likely been provided with reinforcements.
Nkunda and his 4,000 soldiers have been battling with the United Nations-backed Congolese army in North Kivu province since August.
The United States recently urged Nkunda, who defected from the army several years ago, to surrender and go into exile to avoid further violent clashes in the area.
Nkunda formed his own militia after Congo’s war ended in 2002.
He claims to be protecting the minority Tutsi ethnic group from Rwandan Hutu rebels who have occupied forests in east Congo since fleeing Rwanda's 1994 genocide.
More than 500,000 ethnic Tutsis and moderate Hutus were killed during the Rwandan slaughter 13 years ago.
Jendayi Frazer, US assistant secretary of state for African affairs, said in November the best solution would be for Nkunda to go into exile.
The fighting has displaced between 400,000 and 500,000 civilians, according to the UN.