Since you worked in VA, you will need to file a VA non resident tax return to claim the income you earned during those 10 weeks. Whether or not you get a refund depends on how much you made and how much they withheld to cover your tax liability to VA.
It was not a mistake for them to withhold money from your wages to VA as you do have to pay taxes on the income you earn in another state, even if your state does not levy income taxes.
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