Bloomfield has accepted responsibility and apologised on behalf of the ministry.
But the issue came into sharp focus yesterday, after Clark told media: "The director-general has accepted protocol wasn't being followed; he has accepted responsibility for that and has set about putting it right".
He delivered this line while Bloomfield was standing behind him.
A MediaWorks video captured Bloomfield's face while his boss was making the comments – that video has garnered much attention online, with some opining that Clark threw Bloomfield under the bus.
National health spokesman Michael Woodhouse has pointed out that the pair's body language shows they have fallen out – something Bloomfield has denied.
Muller, who has been calling for Clark's resignation for weeks, said the Health Minister's conduct was simply not acceptable.
"While Dr Bloomfield has fronted up day after day, Clark hasn't even bothered to look at the quarantine arrangements that are so vital in protecting New Zealand from the virus."
Muller said that while the Minister of Health's continued "bumbling presence" defines the incompetence of the Labour Government, Bloomfield shouldn't be the one who should accept responsibility for the furore.
During the lockdown, Clark was demoted to the bottom of the Cabinet rankings after he revealed he broke level 4 rules and took his family to the beach.
Prime Minister Jacinda Ardern said she would have sacked him for this if New Zealand was not in the midst of a public health crisis.
Muller said Ardern needed to now take action.
He said she was happy to take centre-stage during lockdown briefings but as soon as there's bad news, she is nowhere to be seen.
"For Ardern, when things go wrong, the buck stops with the frontline workers, never her ministers, never herself."
The clip of Clark and Bloomfield has gone viral on social media.
Many people have been quick to defend Dr Bloomfield while also heavily criticising Clark.
"Bloomfield was fronting every day doing what the Health Minister should be doing. Meanwhile, Clark was swanning around mountain biking," one Twitter user wrote.
"He didn't even follow lockdown himself," another said.
A week into the country's full-scale alert level 4 lockdown in April, a photo of Clark's van - complete with his face plastered on the side of it - was spotted at a Dunedin mountain bike park.
He admitted going for a bike ride and later revealed he had also taken a 20km trip to the beach, also during lockdown.
'Bloomfield has more class than that'
Reacting to the footage, other members of the public labelled Clark's move and failure to take any responsibility for the botch-ups as "spineless and cruel" and called for his resignation - again.
"Dr Bloomfield, the professional that he is, has been front and centre for months speaking for whole response while Clark has been (missing in action). Clark should be fired."
"Ashley Bloomfield fronted up nearly every day of the lockdown - probably didn't even rest on his rare days off," another tweeted.
"He kept us informed, didn't dramatise, just gave us the facts. He could have made mince meat of David Clark over mountain bike-gate, but the man has more class than that."
As of yesterday, New Zealand has 11 active Covid-19 cases all connected to recently arrived overseas passengers.
None of those people are in hospital and all are being quarantined.
