Cant' figure if I have to claim NYS refund on previous tax year on 1040 as additional income?

If you itemized deductions for your 2018 return, then the 2018 refund that you received in 2019 may be taxable.  If you did not itemize deductions, then it is not taxable.

After you enter the information about your state tax refund, there will be some additional questions to answer and some information requested from your 2018 return so that TurboTax can determine if the refund is taxable.  

Use these steps to get started by entering your 1099-G reporting the state tax refund:

• On the top row of the TurboTax online screen, click on Search (or for CD/downloaded TurboTax locate the search box in the upper right corner)
• This opens a box where you can type in “1099-G” (be sure to enter exactly as shown here, including the dash) and click the magnifying glass (or for CD/downloaded TurboTax, click Find)
• The search results will give you an option to “Jump to 1099-G”
• Click on the blue “Jump to 1099-G” link