I don't really like mishear much, because I can't read it, but it means hear incorrectly . Overhear, on the other hand, means to hear a conversation to which one is not a party . So the meanings are different.
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Mister MicawberI cannot locate any dictionary that defines overhear as mishear. Could you give us a specific reference, please?"I misheard you" in the past tense is not uncommon.
I cannot think of a one-word synonym-- we usually say something like, did I hear you correctly?