The Government is asking the general public to refrain from walking up to a vaccination centre for a jab because it might prevent someone with a booking from getting one.

This follows an Auckland family calling the rollout a "shambles" after an 81-year-old was turned away from her pre-booked appointment when the Highbrook Rd site closed due to a lack of space.

This was one of several examples of people showing up to get vaccinated but having to wait for hours and sometimes being turned away.

Multiple Auckland vaccination sites are now trialling walk-in vaccinations, but Covid 19 Response Minister Chris Hipkins is asking those in group four - the general public - to wait until their scheduled rollout from July.

He said what happened at Highbrook was "very unfortunate", and eligible people with a booking should be able to get a jab.

"We've seen some limited examples where there have been challenges. We would like to avoid those sorts of things happening," he said at a press conference today.

"These are early days. They're scaling up quickly. There will be the odd teething problem. If we all do what we can do to help, we can minimise some of that.

"Our priority has to be for those who have booked and are eligible."

He added that DHBs still had their own vaccination plans, and they might including vaccinating everyone in a remote area in one go.

As of midnight last night, the rollout was tracking slightly ahead of forecast, with 120,000 people fully vaccinated and almost 390,000 doses administered so far.

But less than 9 per cent of those with two doses were Māori, while the figure for Pasifika was less than 7 per cent.

Hipkins said that was partly because those ethnic groups were underrepresented among health workers, who are in a priority group to get vaccinated.

Māori and Pasifika health providers would play an important role in lifting those numbers when the general rollout started from July.

"We've got to do better - there's no question about that," Hipkins said.

Director-general of health Ashley Bloomfield added that there were "very specific plans" in place to reach those communities, as well as for parts of the country where vaccine hesitancy was more prevalent.

"There are some locations - Northland is one where the childhood vaccination rates are lower than other areas around the country," Bloomfield said.

"We know there is a particular dynamic on the West Coast and also in Golden Bay."

Hipkins said he didn't think the level of vaccine hesitancy was so high that it would prevent New Zealand from reaching population immunity.

But he couldn't see the borders opening beyond other "safe zones" until next year at the earliest.

He was confident in the larger supplies of the Pfizer vaccine arriving from July, though they were "not 100 per cent locked in".

Hipkins wouldn't be drawn on the decision of Labour MPs on the health select committee to vote down a motion to be able to grill top Covid officials - including Bloomfield - each week.

"That's a matter for the select committee," he said.

Bloomfield said there was no need at this stage to pause the trans-Tasman bubble with Melbourne because the risk was low.

People in Victoria were asked to follow local health advice, and the Ministry of Health were in the process of contacting the 4500 passengers who had recently travelled from Victoria to New Zealand.

If any of them had been at one of the locations of interest, they were legally required to isolate for 14 days after the date of potential exposure.