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Richardson reported the sore throat to cricket Australia's medical staff on Thursday.
Team doctors are confident the issue is not serious and only a normal throat infection, but he has been told to stay away from the squad given his recent return from South Africa.
The news came just an hour before Australia's clash with the Black Caps behind closed doors, with fans barred and the SCG's doors locked.
"Our medical staff are treating this as a typical throat infection," a Cricket Australia spokesman said.
"But we are following Australian Government protocols that require us to keep Kane away from other members of the squad and perform the appropriate tests given he has returned from international travel in the last 14 days.
"Once we receive the results of the tests and Kane recovers in the next few days we expect he will re-join the team.
"We will not be making further comment until something changes."