Hello atv remote,
If you have issues with Siri recognizing you on your HomePod mini, here are some steps to help:
1. Reset Recognize My Voice: In the Home app, tap Home
in the upper-left corner, then tap Home Settings. Tap your name under People, then turn Recognize My Voice off then on. Wait a few minutes before trying Siri again.
2. Restart the iPhone, iPad, or iPod touch that you use with "Hey Siri."
3. Restart your HomePod mini or HomePod.
4. Set up "Hey Siri" again: On your iPhone, iPad, or iPod touch, go to Settings > Siri & Search, then turn Listen for "Hey Siri" off then on, and follow onscreen instructions to teach Siri your voice.
5. On your iPhone, iPad, or iPod touch, go to Settings > Siri & Search, and make sure that it's set to the same language as your HomePod.
You can see these steps here: Set up voice recognition on HomePod mini or HomePod
If you still have issues, and so we can further isolate the cause of this issue, try moving your HomePod mini to another room to see if that helps.
Best Regards.