Can I use screen mirroring on Apple TV app on PS4

The TV app is merely for having access to iTunes purchases and supported streaming services (Apple TV Channels) in the app. In order for one to mirror a device would also need to support airplay 2, which the Playstation does not. So the option would come down to using Apple TV hardware, Roku or supported Smart TV.

Apple TV app - Devices - Apple

and scroll to

AirPlay 2–Enabled TVs and Devices

playstations does not appear to support being mirroring recievers

and no devices by iphone, ipads, iphones and mac computers support being mirroring senders