When you speak about grammatical or orthographical (or any onter) mistakes, it's always исправить. Поправить may be used in the sens "to improuve", to make something that is not completely wrong better.
Поправить ошибку sounds completely wrong to me even if I cannot give you a rule for that ;). You can поправить того, кто ошибся, that is to correct someone's mistake by pointing to them thir error end/or giving them th correct answer. But if in English in this context the action is directed to the mistake, in russian it is directed to the person that made a mistake. So you can say исправить ошибку or поправить человека, сделавшего ошибку. I hope that it is more or less clear...
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Поправить ошибку sounds completely wrong to me even if I cannot give you a rule for that ;). You can поправить того, кто ошибся, that is to correct someone's mistake by pointing to them thir error end/or giving them th correct answer. But if in English in this context the action is directed to the mistake, in russian it is directed to the person that made a mistake. So you can say исправить ошибку or поправить человека, сделавшего ошибку. I hope that it is more or less clear...