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Mage hand doesn't make a hand, therefore there is nothing to effect. Reread the spell description. It's just telekinesis on one object. If your game plays that there's a "special effect" of a hand that appears, that's fine, but has no bearing on the spell effect, which has no such thing.
Since "mage hand" isn't described as a construct of any sort or having any hit points, I'd assume nothing happens. Heck, part of the appeal of Mage Hand is that it nets you fine manipulation at range with little risk to yourself.
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