If the AirPlay icon doesn’t appear, please see this article.If connecting from a public WiFi (like hotel or airport), Airplay usually doesn’t work due to public WiFi security features.Move your HomePod or Apple TV away from other electronics if experiencing stuttering or stoppages.Update iOS, Apple TV, and HomePod software.Reset your network settings on your iDevice in Settings > General > Reset > Reset Network Settings.Disconnect from any other Bluetooth devices (like speakers).Turn off Bluetooth or if already off, try turning it on.Restart all devices and try to AirPlay again.Plug all devices into a wall outlet instead of using battery power.Setup an IP reservation or quasi-Static IP on Apple TV.AirPlay Not Working Troubleshooting tips.Trouble with older devices and your Mac?.
AirPlay versus AirPlay Mirroring–What Gives?.