The comment came as claim and counterclaim continued yesterday between the two Dunedin City Council figures at the heart of the controversy over an $8 million shortfall.
Mr Cull and Cr Hudson, differed again yesterday on details of the issue, including whether Cr Hudson had assured the council the companies could come up with the money.
Yesterday, Cr Hudson also said Mr Cull had visited him at home during his recuperation from an accident earlier this year, and engaged in a "very inappropriate and unfortunate conversation" about DCHL matters.
He declined to explain the detail of the conversation, and Mr Cull was dismissive when asked about it yesterday.
"He should put up or shut up," the mayor said.
He then added: "Given he was recuperating at home, I pursued some discussion in his home rather than him having to come into town."
The conversation never got out of control and was "entirely professional", Mr Cull said.
Since the deficit news broke on Friday, Mr Cull has claimed Cr Hudson should have made it clear sooner DCHL would not be able to pay the sort of dividend the council had included in its budget.
Mr Cull said Cr Hudson had been asked in council meetings whether DCHL could handle the level of those dividends.
"He said yes."
Cr Hudson responded yesterday Mr Cull had "better be specific, and identify the dates".
"I don't believe that I have ever said that."
He said he had been clear "for some time", going back to last year, there was an issue.
Mr Cull responded late yesterday he did not have the dates, but: "I'm not the only one that remembers it".
There were "a number of councillors" who did, and it was "quite clear".
Asked late yesterday about criticism levelled at the board, Cr Hudson said he was "not surprised" Mr Cull was suggesting changes.
"I don't have any difficulty with that, provided the appropriate process is followed. I don't believe the appropriate process is [being] followed.
"If at the time for my reappointment for company boards, [the] council decides they would like me to retire, that's their prerogative ... that's fine, that's life."
Asked if he still saw a role for himself beyond his current appointment, Cr Hudson said: "I'm not sure that I would want it, given, in my view, the inappropriate criticism which is clearly coming through from the mayor's comments."
Cr Hudson also said that it was he, not Mr Cull, who was behind setting up the council-controlled organisation liaison committee that Mr Cull has said got to the bottom of the issue.
As chairman of the DCHL board, he had asked finance, strategy and development committee chairman Cr Syd Brown at the beginning of the year to meet chief executive Bevan Dodd and himself to suggest the idea.
Mr Cull might have formed the liaison group, but Cr Hudson said he had initiated it.
Cr Brown said yesterday there had been discussions, though he was unsure, in the end, whether that was behind the outcome.
He believed the liaison group had come from discussions during annual plan meetings in May, though he said Cr Hudson had certainly not objected to the idea.
Cr Hudson said the liaison group had only "one brief meeting" with the council companies, and had made no attempt to interact with the companies to fully understand the financial position.
Asked to respond to claims in the last few days he had kept voting for spending when he knew the extent of the situation, Cr Hudson said there were "a number of things I didn't vote for because of the knowledge".
"The last one of those is the town hall upgrade," he said. He voted against it last year.
"I was concerned that we were committing to projects that would give no financial return to the council."
He said the council had done "a lot of things that are nice to do".
"I don't regret some of them; they were important to the health and vitality of the city."
Cr Hudson said he had planned to table documents at a non-public council meeting last week, detailing DCHL briefings to councillors about debt levels and dividends.
However, he instead quit the meeting after being challenged over a perceived conflict of interest by another councillor, because of his dual roles, which made him feel "uncomfortable".
"I had no choice but to leave." Cr Hudson would not name the councillor or release the material to the ODT yesterday.
"I'll release it on legal advice at the appropriate time," he said.
