Hardwire a Wi-Fi Extender with an Ethernet Connection

Yes, if the extender has an Ethernet port and supports access point mode. Connect it to the router or network by cable and select AP mode as described in its manual.

Usually no. Repeater mode uses a wireless connection to the router; access point mode uses Ethernet as the backhaul and broadcasts Wi-Fi from the wired location.

It can remove the wireless backhaul link as a bottleneck, but results still depend on the router, port capability, Wi-Fi standard, device and internet service. It does not guarantee a particular speed.

Some models provide Ethernet ports for wired clients, while others reserve a port for backhaul or have different mode limits. Check the model manual for how its port works in AP mode.