Our Principal was walking around in the school, inspecting the kids and see what they were doing(such inspections are always sudden). So our teacher said:
Sit up straight, Ms.X is on a round.
Is the use of "on a round" natural?
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It might or might not be the normal word in a particular school, where they might use "round", "patrol" or "tour". "Round" is certainly the normal word for doctors in a hospital (as is "the rounds")
Your sentence reads perfectly naturally, as do these: