A person who arrived at Middlemore Hospital and then left the grounds after testing positive for Covid-19 is due to be moved to a quarantine facility today.

The patient is one of two people connected to two exposure events at the South Auckland hospital's emergency department this week.

The Ministry of Health reported yesterday that the pair - who are not connected to one another - arrived on Wednesday for unrelated health conditions, but were then tested for the virus.

"The second patient who had self-discharged is being managed by the Auckland Regional Public Health Service and has been isolating at home," the ministry said this afternoon.

"Today they are being moved into a quarantine facility."

Speaking at this afternoon's Covid press conference, Deputy Prime Minister Grant Robertson said the person went straight home before they were contacted by health officials.

Health officials said of the 34 patients who were regarded as close contacts after the pair tested positive for Covid, a further six people have since been discharged from hospital.

"They are being followed up by public health, along with the other 32 outpatients who were considered contacts."

The health ministry acknowledged that the number of exposure events popping up at Middlemore Hospital was not unexpected, given there are Covid sub-clusters in the South Auckland region - for which Middlemore is the local hospital.

"In addition, Middlemore is the receiving hospital for a number of quarantine facilities in the area and as such, regularly Covid-suspect and Covid positive patients from these facilities."

Robertson reiterated this, acknowledging Middlemore Hospital's frontline role during this current Auckland outbreak.

Some cases arrive quite unwell and had to be admitted. There will be some where testing doesn't occur, Robertson said, but that is not a problem of the hospitals, for they are the solution.

"We do also really want to make sure people retain confidence in our hospitals," Robertson said.

"And it's important [people] still go there if they are unwell."