Can I start a desktop return and then work on it on mobile app?

[DoninGA]

No.  The 2023 desktop CD/Download edition on your computer cannot be transferred to the 2023 TurboTax online editions.

[VolvoGirl]

If you mean start an Online return on your Desktop then yes you can switch between the Online version and the mobile app.  The mobile app is just a portal to the Online version.  The actual Desktop program that you install is completely separate from Online and has different prices and features.