Eight children were flown to Australia from Nauru on Monday for medical treatment, refugee advocates say.
The Asylum Seeker Resource Centre said 25 people in total left the country yesterday, with family members joining the children being evacuated.
About 600 refugees, including 27 children remain in exile in Nauru, many of whom are suicidal and suffering from anxiety, the centre said.
Australia's government will not allow anyone evacuated from offshore detention in Nauru to resettle permanently in Australia.
Last week, it said it would bring all the refugee children in Nauru to Australia for treatment by the end of the year.
But the government continued to fight legal applications to evacuate sick children, the centre said.
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.
RNZ Pacific understands about 10 of the men due to be moved have not completed their treatment at the hospital.
Refugee advocates said others had been waiting months without receiving any medical attention.
Some of the refugees have been told they will be returned to Port Moresby in four weeks.
One man attempted suicide on Wednesday after he was told he would be sent back to Manus.