It depends on how large the refund offset was and whether or not your state refund covered the entire refund offset. If you still have a past state/federal debt, overdue child support etc, your federal will also be impacted. 

A refund offset is when an IRS refund is reduced or intercepted to pay off delinquent debt, such as past-due child support, outstanding student loans, or unpaid state income tax.

You'll get an explanatory letter from the Bureau of the Fiscal Service if your refund is offset. The remainder of your refund will be processed as usual; an offset shouldn't delay it, unless of course your entire refund was applied to your debt. The IRS Where's My Refund tool will contain a disclaimer if your refund may have been offset.

More info about refund offsets, including contact info and the appeals process, is on IRS.gov (Tax Topic 203).