Welcome to the Apple Support Communities. I found an article to help troubleshooting AirPlay.
If you continue to have issues, the next step would be restore the Apple TV.
The issue isn't that I cannot connect to AirPlay or the AirPlay icon doesn't show up when intended. My issue is that it keeps showing up and disappearing while the AppleTV is "sleeping."
I don't care that it does it, but the consequence of this frequent "stuttering" is that my audio from anything over my network (audio from anything over the Internet and from my network drive) gets cut off for 1 to 2 seconds.