Your Google Home and Google Home Mini can control any TV that's equipped with Google's Chromecast technology, whether built-in or via an HDMI Chromecast unit. It's not quite as easy as simply getting your control device onto the same Wi-Fi network as your target device and barking commands, but the process is pretty simple and only takes a few minutes. You'll use the Google Home App on your compatible smartphone or tablet to both link up the device in question and any applicable streaming video services. Once you've done all of that, any Google Home units on the same network that are signed into your account can tell the TV what to play at any given moment.
The first steps to setup are to ensure that the control devices you want to use are all on the same Wi-Fi network as the target device, and that they're all signed into the same Google account. From there, go into the Google Home app, tap on the hamburger menu icon in the top left of the screen, and tap on "More settings." From there, under the "Services" heading, you'll find "TVs and Speakers." Tap on that, then tap the plus sign in the new screen that pops up in order to add a new device. If the device you want to control is on the same Wi-Fi network and everything is working right, you'll see it in the menu. Give it a tap, and it's added. The compatibility with streaming services is a bit limited, and you'll have to go into the "Videos and Photos" menu in the same place you found "TVs and Speakers," then tap on "Video." You'll be able to add compatible streaming services from there, and then you can ask your Google Home to have any Chromecast-enabled devices play from them.
If you want to unlink any devices or services from your Google Home, simply go into the same menus where you added them, tap on them, and remove them. If anything goes wrong during adding or removing any devices or services, you can briefly reset your Google Home to try again by unplugging it and plugging it back in. Major problems may require a factory reset of your Google Home, which can be done by holding the microphone mute button on the unit for 15 seconds, then confirming that you want to reset it.