Refugee advocates say eight children were flown to Australia from Nauru for medical treatment on Monday.
The Asylum Seeker Resource Centre says 25 people in total left the country yesterday, as family members joined the children being evacuated.
The centre said among about 600 refugees, 27 children remained in exile in Nauru some of whom remain suicidal or suffer from anxiety.
Australia's government said it wouldn't allow anyone evacuated from offshore detention on Nauru to resettle permanently in Australia.
It said it would bring all the refugee children on Nauru to Australia for treatment by the end of the year, but the centre said the government continued to fight legal applications to evacuate sick children.
Meanwhile, authorities in Papua New Guinea have flown more than 20 refugees from Port Moresby to Manus Island to make way for the APEC summit.
About 50 refugees, staying in a motel in the capital while they receive medical treatment, have been told they'll be returned to the island before APEC.
On Friday and Saturday, 23 of the refugees were forced by police from the motel in the early hours of the morning.
They have been moved to relieve pressure on Pacific International Hospital in case APEC delegates and conference employees need treatment