The government refuses to distribute the cost of living payments to those with zero taxable income, including up to 70,000 students.
A statement supplied by Gayford said an NZME podcast and social media page made "untrue and damaging” about him.
The climate change minister cited Budget timing and recent climate-related investments as reasons for the change.
Michael Wood is open to making changes to the green list for immigration before the scheduled review next year.
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