The person misunderstood in public may need the clarification to travel just as publicly.

Repair usually has to reach as far as the harm did. A private apology after a public misunderstanding leaves somebody corrected in front of everyone and cleared in front of one person, which is not an equal exchange. Setting the record straight where it was set is uncomfortable in a way that a quiet word is not, and that discomfort is roughly the size of what was taken. Being right in private is only half of being cleared.

If you got something wrong about somebody in front of others, put the correction where the mistake was. Say it without explaining yourself at length.