The Prime Minister said there had been complicating factors including "language barriers" when attempting to contact trace the latest people found with Covid-19.
But the Auckland Regional Public Health Services (ARPHS) said that prior to this outbreak, it had not experienced any issues with language barriers.
Jacinda Ardern confirmed the two people who tested positive for Covid-19 on Saturday caught the virus after one of them visited another family during level 3 restrictions.
Under level 3, you may only leave your home for essential purposes, such as visiting the supermarket or pharmacy.
ARPHS also would not elaborate on what language barrier issues were faced with contact tracing the families of the two positive cases.
"ARPHS uses professional interpreting services as required when engaging with people affected by Covid-19 - cases and contacts," a spokeswoman said.
"The service also makes translated information available via its website. Those we speak with are asked, at the outset, what their preferred language is, and we arrange the appropriate interpreter if needed."
The Ministry of Health said diverse groups of people that live in New Zealand required tailored communications and approaches to meet their unique needs.
"Translation of Covid-19 information is undertaken at an All of Government level into more than 20 languages, plus accessible formats for the disabled community, available from the Unite Against Covid website," a spokeswoman said.
The translation services were generally provided by translators at the Internal Affairs and Te Taura Whiri i te Reo Māori.
The spokeswoman says the ministry also works with Cause Collective, Pacific Media Network and Ministry for Pacific Peoples to communicate with Pacific communities, as well as church and community leaders when appropriate.
"To ensure the ministry's campaigns resonate with Māori, it partners with iwi communications professionals and Māori creative agencies, such as Mahi Tahi," she said.
These groups provide insights and guidance on how to best reach Māori communities, she said.
Asian communities were reached in several ways, with messaging sent directly to East Asian media outlets which she said were closely linked to their communities.
"We actively engage with Asian communities ... all information and messaging in support of our response to Covid-19 is translated into 23 languages. Of those seven are East Asian," she said.
"Ethnic media respond very positively to translated content that is made up of short messaging – they can publish it immediately on news websites, and especially in the case of Chinese media and community groups, on their Weibo social media profiles."
Profiles highlighting the work of individuals working to keep their communities safe are published regularly on https://covid19.govt.nz/everyday-life/support-your-community/community-heroes/
The spokeswoman said ARPHS had no issues accessing support for Auckland's different ethnic communities.
"ARPHS also makes translated information available for cases and contacts in a range of languages on its website," she added.