How to file FTB3853 to apply for the short gap coverage exemption using FreeFile? FTB said I need to enter code "C" in section III of this form.

Yes, TurboTax will fill in the coverage gap exception, but you can't select any of the exemptions. You can revisit your California tax return and change your answers in the health coverage section. Here's how to do that:
 

1. Sign in to your TurboTax account.
2. Select State Taxes, then Edit.
3. Select Your State Returns.
4. Select Edit to the right of California. 
5. Continue until you get to Your 2023 California taxes are ready for us to check, select Edit to the right of General info.
6. Continue until you get to Did you have health insurance coverage in 2023?
7. Answer according to your situation.
8. On the screen Select the months you had health insurance in 2023, check the box next to each month you had insurance, then select Continue.
9. On the screen, Did you have a situation that waives the tax penalty for being uninsured? don't check any of the boxes. 
10. Select Continue. 
11. If your gap qualifies as a short gap, you will see a message saying the penalty has been waived.