The person was transferred to the Auckland quarantine facility on February 19 as a precaution.
Due to Case H being in isolation while infectious there were no additional contacts to report, said the ministry.
Because of steps already taken in identifying, testing and tracing individuals linked to the February cases, as well as Case H isolating at home since Monday and then being in quarantine for the last two days, the public health risk was considered very low, said the ministry.
A decision on whether Auckland will remain at alert level 2 or move down to level 1 will be made by Cabinet this afternoon. The Prime Minister is expected to announce the decision at 3pm.
Meanwhile, some casual plus contacts at Papatoetoe High School have been required to have a second PCR test. It included four teachers and 28 students where a record of the test result wasn't available or a more recent test was required.
Close contacts were expected to return to school on Wednesday, February 24 or when advised by Auckland Regional Public Health Service.
The ministry said contact tracing had identified a total of 125 close contacts associated with all cases in this cluster, aside from the positive cases which have previously been reported.
Three previously reported close contacts have been ruled out on further investigation and one infant is not required to be tested.
Of the remainder, 122 of the close contacts have returned a negative test result.
"We are awaiting test results for two people - all of whom are from the medical clinic and relate to Case C, which is considered a low risk exposure event," said the ministry.
As at 8am a total of 31 close contacts and 1416 casual plus contacts have been identified at Papatoetoe High School. The number of casual plus contacts had increased by three following further investigation.
Of the casual plus contacts comprising other students and staff at the school, 1402 have returned negative results. There was one positive (Case E), and 13 results were still outstanding.
"Once again we would like to thank the teachers, staff, students and their families for testing and isolating as required to keep their community safe," said the ministry.
The ministry said the Auckland Airport precinct where the woman at the start of the February cluster worked remained "the most likely source of the outbreak", but further testing has not identified any potential transmission routes.
All scenarios for possible infection sources of the Auckland February cases continue to be thoroughly investigated.
ESR has been reviewing all B.1.1.7 strains identified over the last two months to see if there was any possible link to the latest case.
The Four Points by Sheraton managed isolation facility in Auckland continued to be investigated.
This was where there is a possible genomic similarity between a previous positive person who had since recovered and the current community cases, though noting it was not a direct match.
There were 265 returnees at the facility in late December. Thirty-six of these were based overseas. At this stage just 11 out of 229 based in New Zealand have not been contacted. "We will continue to follow up with these," said the ministry.
"It is important to note that health officials consider this an unlikely source of the infection at this stage but are pursuing it as part of actively chasing down every line of enquiry," said the ministry.
Meanwhile, the case in managed isolation reported yesterday is now on board a commercial vessel at sea.
The mariner tested negative on day one in managed isolation and then tested positive on day three. The result of this second test came in after the crew member boarded the vessel. "We are now awaiting genome testing results," said the ministry.
The case had no contact with port workers and given that all policies and procedures were followed the risk to the community is considered very low.
The ministry said mariners are allowed to leave managed isolation and board vessels, before a test result is known to limit disruption to international shipping.
Health authorities remain in contact with the vessel about appropriate quarantine arrangements for all crew on board.
Of the latest cases at the border, two people one person arrived on February 20 from Qatar and tested positive in routine day 0/1 testing. A second person travelled from Iraq and also tested positive in routine testing on arrival.
Two people who arrived on February 18 from USA and Colombia and in managed isolation in Christchurch each tested positive during day 0/1 routine testing.
Another who arrived from Morocco on February 17, also tested positive for the infection on arrival and was in a Christchurch facility.
The ministry said all were deemed historical and not infectious.
One further case dating back to December 17 was reported today. The person arrived from the USA and tested positive on routine day three testing. The ministry said the delay in reporting this case was due to checking it had not been previously reported elsewhere.
There are now 54 active covid cases in New Zealand.
We have now passed the 2000 mark with 2001 confirmed cases of covid in New Zealand since the outbreak started.
After a week of intense testing with saw more than 71,000 tests processed since last Sunday, only 3882 tests were processed yesterday.
Since the start of the year there have now been 33 historical cases out of a total of 190 cases.
The Covid tracer app use continues to grow. There are now 2,671,646 registered users, with an increase of more than 110,000 users in the last fortnight.
A fresh plea was issued for Kiwis to keep a record of their movements. "It's critical to keep track of where you've been and the Covid Tracer app is an easy way to do this. Please continue to scan QR codes wherever you go and turn on Bluetooth tracing in the app dashboard if you haven't already done so," said the ministry.
As the vaccine campaign begins, Cabinet was due to decide this afternoon whether Auckland should have its alert level settings revised.