Connect home Wi-Fi router to mobile hotspot?

Should be possible.
Either by setting up your router as a repeater to your phone's WiFi (if the devices of your home network are all wireless) or setting it up as a WiFi-bridge (if you want to connect devices via ethernet to the router). Most modern routers have these capabilities.
If yours doesn't (if you can't find it in the settings, maybe consult the manual), you might be able to flash an alternate firmware, like DD-WRT, which provides these capabilities (check the DD-WRT database to see if your router is supported: https://dd-wrt.com/support/router-database/ )
If you decide to flash a different firmware, be sure to read up on the process, so you understand what you're doing.

 

Even if possible, it is almost definitely a violation of your carrier's terms of service and could get your service terminated for excessive data use. Also the cell network may not have the bandwidth or stability to be a good substitute for a proper ISP service.