He also expressed fear about the possibility of the reintroduction of a stricter lockdown, saying it would kill jobs and cause the "humanitarian disaster" to worsen.
Mkhize visited the Eastern Cape last week to assess efforts to curb the transmission of the virus in the province.
As of Monday, positive Covid-19 cases in the province had spiked to 126,520.
Speaking at Nelson Mandela University in Port Elizabeth, the minister said the department was concerned about the increase in infections in Nelson Mandela Bay.
“We are here because there is clearly a resurgence in Nelson Mandela Bay. It is a concern we are also seeing in a few other areas. The country has small cluster outbreaks, but they are transient. Nelson Mandela Bay has continued to fester and our concern is that something has to be done,” he said.