Hi ThomasM2,
We understand you're experiencing a couple of issues with Handoff and Siri. There are a few steps we would recommend.
First, ensure that the HomePod mini and iOS device are up to date:
Update your iPhone, iPad, or iPod touch.
Update HomePod
Next, check that the Apple ID signed in for iCloud on the iOS device is the same Apple ID set as the default account for HomePod
If the Apple IDs are different, make sure the iCloud Apple ID has been invited to the HomeKit setup that the HomePod is a part of:
Share control of your home
You'll also want to confirm iCloud Keychain is enabled for the Apple ID:
Set up iCloud Keychain
In addition, make sure the iOS device is awake and unlocked when you attempt to initiate audio handoff. Are the iOS device and HomePod are on the same Wi-Fi network? We ask as some routers broadcast two networks - 2.4GHz and 5GHz. The iOS device and HomePod have to be on exactly the same network.
Once all of the above is confirmed, remove any protective casing from the iOS device. Restart the router, the HomePod, and iOS device and test again. Then test Audio Handoff using a Phone call:
Use HomePod for phone calls
And lastly, try changing Speaker & TV settings in Home Settings to Everyone
Hope this helps!
Have a great day.