You say that the card works on Windows. Are to 100% certain that in TVH you have your muxes configured correctly? Are you in exactly the same location using the same signal cable for both systems?
If the problem is missing firmware, perhaps you could look in dmesg
to see if there is any mention of missing firmware when the kernel detects the device at bootup. Perhaps you could compare this to your Xubuntu/Kaffeine dmesg
to look for any clues.
If there is a message, it could mention the device by name, or, if the card is assembled from a generic chip set, perhaps that chip set would be mentioned instead.
If you don't mind installing other operating systems on a spare hard drive, you could try installing LibreELEC and see if that works. It is a bundle of Linux + TVH + Kodi that can be used as a multimedia appliance. The important thing is that LibreELEC has a lot of firmware files pre-installed. You may get lucky.