HomePod Mini randomly cuts out when using as speaker with Apple TV 4K

Start with the basics (which you may have already done).

• Reboot the HomePod Mini
• Is the HomePod Mini up to date OS?
• Try it from several different apps, is it happening with one particular app? See if you can pinpoint it to specific conditions the audio cuts out.
• Reseat the HDMI cable, try a different cable (unlikely but I always suggest eliminating the basics even if unlikely).
• What if you test by routing the audio to the TV? Do you get audio cutout with TV speaker audio? Is it just HomePod?
• I seem to recall a Support thread regarding neon lamps nearby the HomePod. I cannot find it to give credit to their poster but I'd consider that: what electrical device is very close to your HomePod, an older LED bulb perhaps, turn off all those and test. I'd even consider what else is plugged in on the same electrical circuit.
• Here is a separate support post regarding router configurations and what HomePod likes best. Airplay can definitely be impacted by Wifi nuances: https://discussions.apple.com/thread/252548018