backing up an external NAS hard drive

Time Machine was designed to back up your Mac and a "local" hard drive......like a USB drive connected directly to your Mac. Time Machine cannot back up a network drive.

You can use another application like Carbon Copy Cloner, SuperDuper or ChronoSync to back up the network drive to another drive. Personally, Carbon Copy Cloner (CCC) has been flawless for me over many years.

+1 for CCC as it has worked great for me. No pecuniary interest.

3 other hard drives connected as NAS (network attached storage) drives. Those are connected to my network router, containing all my music and some photo data

What make and model router are you using?

Once you start having so many drives plugged into your router it is well worth considering a real NAS. You may well be trying to avoid that option due to cost.

Synology is the one we recommend here because it has very good Mac compatibility and is reliable using it with Time Machine. If new prices are two high a second hand 2 bay from about 2018 can be a good option. Or look at alternatives from various linux / BSD community projects.

Any NAS will include its own backup app and you can simply use a USB drive on the NAS itself.. all setup to automatically update as required.

Another way to think about this is to remember that in order to back up using Time Machine, there must be a Mac operating system installed on the device that will be backing up.