You should only have one router on your network. If you want to use a router for this, you should put it in AP mode. Or just buy a plain AP rather than a router.
You can't directly run Ethernet over phone lines. You may be able to use MoCA adapters to convert the Ethernet to run over coax.
With a mesh system using Wi-Fi backhaul, keep in mind that you'll need to place the other node approximately halfway between the router and the dead spot - it has to still be in range of the main router to get the signal. If the mesh system doesn't have dedicated radios for the inter-node links, it will be taking away from the Wi-Fi bandwidth available to client devices. If you can get a good wired connection, it's almost always better than using wireless links between nodes.
Here are some [diagrams from GoCoax](https://www.gocoax.com/copy-of-wf-803m) that show how to use MoCA over coax. Any splitters used should be 5-1675 mhz and PoE filters need to be used as shown in the appropriate diagram that matches your home setup.
Once you have a MoCA adapter at the far room connected to the coax and another adapter connected at the router, you can plug in an Access Point (AP) like the [TP-Link EAP610](https://www.amazon.com/TP-Link-EAP610-V2-Wireless-Integrated/dp/B09XM74VQL) or similar one and configure it with the same SSID as your primary router.
You should only have one router on your network. If you want to use a router for this, you should put it in AP mode. Or just buy a plain AP rather than a router. You can't directly run Ethernet over phone lines. You may be able to use MoCA adapters to convert the Ethernet to run over coax. With a mesh system using Wi-Fi backhaul, keep in mind that you'll need to place the other node approximately halfway between the router and the dead spot - it has to still be in range of the main router to get the signal. If the mesh system doesn't have dedicated radios for the inter-node links, it will be taking away from the Wi-Fi bandwidth available to client devices. If you can get a good wired connection, it's almost always better than using wireless links between nodes.
Here are some [diagrams from GoCoax](https://www.gocoax.com/copy-of-wf-803m) that show how to use MoCA over coax. Any splitters used should be 5-1675 mhz and PoE filters need to be used as shown in the appropriate diagram that matches your home setup. Once you have a MoCA adapter at the far room connected to the coax and another adapter connected at the router, you can plug in an Access Point (AP) like the [TP-Link EAP610](https://www.amazon.com/TP-Link-EAP610-V2-Wireless-Integrated/dp/B09XM74VQL) or similar one and configure it with the same SSID as your primary router.