Can I add in 2023 (prior year) carryovers to TurboTax if that return was not processed in TurboTax? If so, how is that accomplished?

Yes.  What kind of carryovers?

Enter a Capital Loss Carryover under

Federal Taxes or Personal (Home & Business)

Wages and Income

Then scroll down to Investment Income

Capital Loss Carryovers - Click the Start or Update button

You enter the full amount that carried over, not just the 3,000

For a capital loss carryover -

Click on Federal Taxes (Personal using Home and Business)
Click on Wages and Income (Personal Income using Home and Business)
Click on I'll choose what I work on (if shown)
Under Investment Income
On Capital Loss Carryover, click the start or update button

Or enter capital loss carryover in the Search box located in the upper right of the program screen. Click on Jump to capital loss carryover