It is unlikely that you need to adjust anything. This is why you use a tax software product like TurboTax.
When you entered the state tax refund into the federal interview in TurboTax, TurboTax adjusted the federal AGI. It is likely that TurboTax adjusted the amounts carried to the state at the same time.
Do you have any reason to suspect that it did not?
Please look at your IL-1040 on line 6 - your Illinois tax refund should be there.
If, however, this refund was not from Illinois but from another state, then go back to your Illinois interview in TurboTax, and proceed to the screen headed by "Here's income that Illinois handles differently". Scroll down to "Miscellaneous" and click Start or Revisit for "Other state tax refunds deducted". In this part of the interview, you can enter can enter tax refunds from other states.
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