Yes, you can amend your federal return to include the state refund. But, you will want to wait to amend until your federal return has been accepted by the IRS. (If your return gets rejected by the IRS, then you can add the state refund income and resubmit the e-file.)
Note that amended returns must be printed, signed and mailed as amended returns cannot be e-filed.
Here are instructions on how to amend your return in TurboTax:
https://ttlc.intuit.com/replies/6603004