Malcolm Turnbull has been urged to show he is different from his predecessor and allow New Zealand to resettle refugees from the Australian-run immigration detention, in an open letter from refugees living on Nauru.

The signatories, who also penned a request to the New Zealand government earlier this month, promised they would not use potential New Zealand citizenship to get to Australia, a country they no longer believe cares about human rights.

In 2013 the New Zealand prime minister, John Key, and his then counterpart Julia Gillard struck a deal for New Zealand to resettle 150 people a year from Manus Island or Nauru immigration detention, but it was effectively scrapped by Tony Abbott after he won the federal election later that year.

“Now we are turning to you because it is said we cannot go to New Zealand because your government scrapped the deal,” the refugees wrote to the Australian prime minister this week.

“This is your chance to act now and put us out of this misery. Please talk to the New Zealand government and allow them to take us.”

In response to the first letter, the New Zealand immigration minister, Michael Woodhouse, told Guardian Australia the offer technically remained on the table as part of government policy, if Australia decided to take it up.

The latest correspondence from the refugees, who have been found to have legitimate claims of asylum and had been granted refugee status, urges Turnbull to demonstrate he is “different” to his predecessor.

A letter written by refugees on Nauru to the Australian prime minister, Malcolm Turnbull, saying they have been ‘treated with cruelty and contempt’ and asking for Turnbull’s assistance in convincing New Zealand to resettle them.

Turnbull has said resettlement in New Zealand would be an incentive for asylum seekers to board boats. After five years in the country a refugee can apply to become a New Zealand citizen, granting them work and travel rights in Australia.

The refugees claim that will not happen.

“If we were settled in a country like New Zealand which respects human rights and allowed us to build up a future there, why would we want to uproot ourselves again to come to Australia,” the letter says.

When you don’t have a proper mental state there is no difference between heaven and hell Mehdi

Despite their refugee status they are only able to remain on Nauru temporarily and must find a third country to take them.

The office of the prime minister has been contacted for comment.

One of the letter’s signatories told Guardian Australia the group of refugees had no interest in coming to Australia anymore. “They do not trust Australian government because they have heard many lies from them,” said the man, known as Mehdi.

He said those who procured travel documents after getting refugee status feared the documents were illegitimate and were a “game” provided only in response to international pressure over Australia’s offshore detention network.

More than 800 people have been found to be refugees on Nauru and are living either in the detention centre or in the Nauruan community. Another 543, including 70 children, are awaiting their determinations.

A number of incidents have indicated serious concerns with the arrangement, including alleged assaults of refugees and asylum seekers by Nauruans and other refugees. No one has ever been charged.

Documents from the International Health and Medical Service, the company contracted to run health services in detention, reveal the extent of the serious mental health crisis among detainees. Among the documents is data which shows that the percentage of children’s presentations to a doctor which then result in a mental health diagnosis have double in the past year.

“The one thing we are losing easily is our mental state,” said Mehdi. “[It] is something that cannot be restored easily and its impact can be permanent. When you don’t have a proper mental state there is no difference between heaven and hell.”