In a statement, Mr Clark claimed his health was "fine" and that he was taken out of context at a prizegiving for volunteer firefighters.
He said he needed to clear up what he said at the United Fire Brigade Association in March.
However, in a follow-up statement, he said that "having spent some time over the weekend considering the impact of the code of conduct on me, Karen and our family, and the wider community, I've decided to take a further month off work".
"With reflection, I came back to work too early, after open heart surgery."
The council had approved leave of absence for three months after the surgery, and he had returned to work after two.
"I feel I should use the remaining month to refresh and look more positively to my role as mayor."
He said deputy mayor Tom Campbell would fill the role in the interim, with support from the committee chairmen.
Mr Clark said he expected to return on July 21.
He was earlier reported as refusing to abide by a resolution from councillors which would have prevented him from undertaking ceremonial duties.
Included in the allegations were claims he called volunteer firefighters second-class citizens, verbally attacked the MC and disparaged young people in authority.
Mr Clark said there were two levels to the complaint that emerged from the prizegiving: what was said in the "table banter" before his formal speech and in the opening of the speech, and what was said in the formal speech.
"The main concern seems to be focused on what is claimed to be disrespectful and disparaging comments by me regarding volunteer firefighters.
"What I said was ‘there is some perception in the community that volunteers are seen as second-class to the career/paid firefighters’.
"I do not support that perception and I'm aware that some brigades have issues regarding training, uniforms and equipment and that it was important to support those concerns."
Mr Clark acknowledged he did not provide his side of the story when the code of conduct complaint was lodged against him.
His statement did not address the nature of the "banter" that led to further complaints.
"I stated the moment I was aware of the complaint which was to be processed via our Invercargill City Council code of conduct, I advised management — and later the appointed investigatorv — that the code was not an appropriate mechanism for public complaints, as to do so would quieten elected members from saying anything controversial.
"So, it would have been farcical for me to be opposed to this use of the code and then participate."
Mr Clark raised concern about the "weaponising codes of conduct to restrict free speech in a democracy".
At Friday’s extraordinary meeting, councillors passed a motion which read: "Since council has received advice from the mayor that he may suffer from brain fade for the next two years, that council request the mayor delegate the attendance at all formal external events to the deputy mayor or another councillor for the balance of the triennium".
Mr Clark’s statement also took issue with this.
"My colleagues have also focused in on my medical recovery, some implying I may not be capable of performing as a mayor.
"I've experienced some ‘brain fade’ which is an impact of open heart surgery and could result in some minor disjointed presentation.
"I don't have any current, ongoing impacts and I feel fine."
Mr Clark said he had had a medical test to ensure no memory deterioration.
