"We have suffered unimaginable cruelty and lost any hope... We do not believe you are concerned when our children are assaulted or our women raped because if you were then you would have acted."
The Nauru detention centre has been plagued by allegations of abuse by guards and poor conditions.
The refugees said: "We need a country where we can be safe and have [a] future... Please talk to the New Zealand government and allow them to take us."
Earlier this year, the Nauru refugees wrote to Prime Minister John Key to ask to be resettled in New Zealand.
Mr Key signed a deal with former Australian Prime Minister Julia Gillard in 2013 to take 150 refugees a year from Australia's offshore detention centres.
So far, Australia has not taken up that offer.
The Telegraph reported that Mr Turnbull was reluctant to send the refugees to New Zealand because it could encourage more boat people.
Mr Woodhouse said it was up to Australia to fill the 150 places and they had not done so.
"As such, the places are reallocated to the annual quota, and most recently the places were given to Syrian refugees."
The latest group of Syrian refugees, who are fleeing civil war, began arriving in New Zealand this week.