Napier Mayor Bill Dalton, however, defended the chief executive, saying the call for his resignation was "beyond unfounded, it's ridiculous".
Mr Dalton said he treated Watchdog! with "the utter contempt they deserve", adding their belief they could call for Mr Jack's resignation was "just plain ridiculous".
Mr Jack was a "hardworking chief executive", Mr Dalton said, who had "put his heart and soul into the role".
"Napier City Council chose Wayne Jack from a number, a large number, of candidates because he came to us as a change manager," he said.
"We employed him because we knew we needed to change.
"He has brought change to Napier City Council, and change for the good."
In a statement calling for Mr Jack to resign, Ms Maxwell said she believed the council had not been upfront regarding the poor management of its pound.
The pound came under public scrutiny late last year when a letter from four Animal Control Officers to Mr Jack expressing their concerns about the pound was publicised.
Watchdog! also made claims about the pound's conditions, but their request that council commission an independent review was not progressed by council.
Ms Maxwell stated the group then "took the matter out of [Mr Jack's] hands", lodging a complaint asking MPI to investigate alleged breaches of the Animal Welfare Act.
"Thank goodness we did," Ms Maxwell said.
"In the first of two reports, MPI has found a raft of shortcomings and made urgent recommendations for improvements."
Last month MPI released an "inspection" report to the council focusing on pound facilities and any compliance issues with animal welfare legislation/requirements.
An "investigation" report is expected to be completed in the next few weeks, which will deal with specific allegations of breaches of animal welfare legislation.
The council issued a press release on the inspection report stating it found there were no breaches of the Animal Welfare Code at the pound.
Ms Maxwell stated it had been "sickening" to read comments in the release attributed to Mr Jack.
Mr Jack declined to comment.
Yesterday Mr Dalton said it was "pathetic" that Hawke's Bay Today was publishing the group's call for Mr Jack to resign.
"I think its pathetic the Hawke's Bay Today is still trying to beat up a story that's old news and has no substance whatsoever," he said.
In their statement, Watchdog! also cited remarks from the council's staff association, that "under Wayne Jack's leadership staff morale had never been lower".
Last year, council staff and the staff association told Hawke's Bay Today a restructure at the council had brought staff morale to an "all-time low".
In June, council staff were advised of a proposed new structure which would disestablish 78 roles to make way for 78 new ones.
As well as redundancies, in October Hawke's Bay Today was told the restructure had led to uncertainty among staff, low-morale at council, and the resignations of a number of experienced staff.
