You only need to enter the winnings on the federal return. TurboTax will transfer that income to your state return(s). If you entered the winnings somewhere else, delete that entry. Prepare your nonresident (NY) return before you prepare the Illinois return. Your resident state will tax you on all of your income, but they will also give you credit for any tax paid to NY.
How to delete forms in TurboTax Online
How to delete forms in TurboTax Desktop
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