Can my Apple TV be wired and not wi-fi to work with Homepod?

There any many different routers, all of which have different configuration screens. You can add the Wi-Fi credentials to your Apple TV and unplug its cable to see if helps with your HomePod connection issue, although the first thing I would try would be to power cycle the Apple TV, HomePod, and router, then wait for them all to connect up again and see it it works out.

tt2

If your network is configured correctly it shouldn't matter whether you use a wired or wireless connection. The key is that you're not using AP/client isolation so all of the devices on your network are able to communicate with each other.

tt2