South Auckland NZ First MP Asenati Lole-Taylor, who visited the family last week, said they had been "given the run-around" by a department that was supposed to help families in need.
Tenants Protection Association manager Angela Maynard said the case was "typical of the current climate at Housing NZ" since the agency said in 2011 it would concentrate on its role as a landlord and would "no longer have a role to assist individuals with their wider social needs".
The Gapes family live on an invalid's benefit. Mr Gapes, 40, was brain-damaged when he was hit by a truck at 3. Mrs Gapes, 45, said she had also had "a memory like a sieve" since she received an electric shock, also at 3.
Their eldest child Sophia, 9, is unable to speak more than an occasional intelligible word. Shiloh, 7, and Sarah, 6, have also been diagnosed with developmental delay and/or autism.
Their former landlord at Weymouth, Destiny Church elder Richard Lewis, said he had to install a cage in the kitchen to stop the children touching hot elements, and when they left in January the house needed extensive repairs. "We had to do a full recarpet and repaint and all the fixtures replaced.
"There was damage requiring replastering and repainting all the way through. We had to do a full replace of furniture based on irreparable staining and damage," he said.
Mrs Gapes said the Lewis family took them in and gave them "awesome" support, but in the end they reached a mutual agreement to leave. The family applied for a state house in January, but needed contact details for two support people.
"They told us that if they granted an HNZ home, and that if our children made a mess of their HNZ property, they would kick us out."
Housing NZ Auckland tenancy services manager Denise Fink said "it became apparent a co-ordinated plan needed to be developed to support their very complicated needs".
She said Housing NZ would initiate a Strengthening Families meeting to bring together all the agencies supporting the family.
"While there is no requirement for any applicant for a Housing NZ property to agree to do something like Strengthening Families, and it does not affect eligibility for a state house and does not delay the process, we know with our previous dealings with this family, something like Strengthening Families will achieve the best and most sustainable outcome."