Opposition leader Simon Bridges says Treasury Secretary Gabriel Makhlouf must go.
Problem: Makhlouf is already due to depart on June 27 (when his term expires) and take up a new role with Ireland's Central Bank on September 1.
Many have called for Makhlouf's head, but can someone be pressured to resign - or if the State Services Commission inquiry finds wrongdoing, even fired - when they've already leaving?
Chapman Tripp employment specialist Marie Wisker says you can still be sacked if you've handed in your notice.
She adds that you can't "resign twice" - though with the caveat that "a resignation is simply a notice of an intention to leave at a certain date in the future. It would be open for an employer and employee to have a discussion about whether that date might be brought forward."
A second employment lawyer, Jennifer Mills, backs that take.
And she adds a potential complication for the State Services Commission, which is investigating both whether Treasury's website was hacked and Maklhouf's statements in the days after the incident.
"It would be in all parties' interests to progress the investigation promptly. However, this investigation may continue even after Maklouf has finished his employment in June - and once an employment relationship has ended, there is no longer any obligation for an employee to continue to participate or cooperate in an employment investigation."
That said, Mills notes some politicians in Ireland are calling for Makhlouf's Central Bank appointment to be put on hold until the State Services Commission investigations here are wrapped up.
As such, "it would make sense for Maklouf to continue participating in the investigation and to ensure he has an opportunity to provide his input," Mills says.
Mills and Wisker both refused to offer a verdict on whether Ireland's Central Bank could withdraw its job offer.
But Wisker does say, "If it was a New Zealand governed contract, the offer could only be withdrawn if it had not been accepted by the candidate.
"Once an offer has been accepted, there is a binding contract that can only be terminated in accordance with its terms.
"Under New Zealand law, that would either require evidence of conduct that would either amount to a breach or otherwise conduct that would entitle termination - for example, serious misconduct or conduct that may otherwise bring the employer into disrepute."