There was no evidence to date that BA.4 was more transmissible or causes more severe disease than other Omicron lineages, the ministry said.
There are 466 people in hospital, including 16 in intensive care.
The six deaths reported today included people who had died over the past three days.
Three of the dead were in the 70s and three in their 80s.
Of the deaths reported today, one person was from Northland, one from Taranaki, two from Waikato, and two from Canterbury.
The total number of publicly reported deaths with Covid-19 is now 750, and the seven-day rolling average of reported deaths is 11.
The seven-day rolling average of case numbers is now 7414 – down from 8435 last Sunday.
The regional breakdown of the 466 people in hospital is: Northland 29, Waitematā 75, Counties Manukau 64, Auckland 97, Waikato 37, Bay of Plenty 19, Lakes 2, Tairāwhiti 1, Hawke's Bay 11, Taranaki 7, Whanganui 3, MidCentral 5, Wairarapa 3, Hutt Valley 6, Capital and Coast 12, Nelson Marlborough 6, Canterbury 60, South Canterbury 3, West Coast 1, Southern 25.
The release of today's data came after seven new Covid deaths and 7043 community cases were reported on Saturday.
On Saturday there were 468 people in hospital with the virus, 15 of them in intensive care.
This was down from the highs of 1016 people in hospital on March 22 and 33 people in ICU on March 21.
The deaths took the total number of publicly reported deaths with Covid-19 to 744. The seven-day rolling average of reported deaths was 11.
In terms of case numbers, Saturday's seven-day rolling average of 7413 cases continued a steady decline from the peak of 20,467 on March 8.
However, the decline has slowed somewhat, the seven-day rolling average only dropping from just over 8000 two weeks ago.
While the overall case trend is declining, parts of the South Island, including the West Coast and South Canterbury, are experiencing an uptick.
Earlier today, Associate Health Minister Ayesha Verrall announced new, more accurate Covid Lamp tests, which return results in 30 minutes, will be trialled at Auckland Airport.
The Lamp (loop-mediated isothermal amplification) tests can be self-administered like an RAT (rapid antigen test), but are much more accurate.
They will be trialled at the airport, initially limited to 30 Air NZ staff, and could eventually be used in hospitals, aged care and other sectors.
The announcement was made as New Zealand's border fully reopens for the first time in more than two years to international visitors from visa waiver countries at midnight tonight.