Use the control in the viewing app
After finding your film, open the viewing option you intend to use. If that app supports Google Cast, select its Cast control and choose your receiver. The sending device and receiver normally need to be on the same Wi-Fi network. Check the app's support rather than expecting a Cast button on every movie-information page.
Google's casting instructions explain the supported-app and network requirements. If the receiver is missing from the list, check those connections first. Replacing the movie query will not fix a receiver that the app cannot see.
AirPlay is a separate method
Use an Apple TV or a compatible AirPlay receiver with an app that supports the feature. Apple's AirPlay guidance also explains screen mirroring and notes that some video apps do not support AirPlay. Where available, the service's native TV app may be an easier alternative.
Do not confuse mirroring with casting
Mirroring shows the device's screen, which can include menus and notifications. Casting support can instead let the receiver handle the viewing session. The behavior depends on the method and app. Test audio and captions after connecting, and keep the exact episode selected before you move away from the phone.