I came across this on eBay: BT Mini Wi-Fi Home Hotspot Broadband 600 Kit Powerline Adapter. Would I have to connect my router to my broadband socket as well as this? What about the similar devices that are out there?
They are just the BT version of the devices I recommended on your other thread.
https://www.tp-link.com/uk/home-networking/powerline/tl-wpa4220-tkit/
There's a picture there that indicates that the router doesn't need to be connected to a broadband or telephone socket, and there are manuals for the BT devices that indicate the same, but I need to be sure that is the case. I know it may seem obvious to most, but it doesn't necessarily mean what it suggests.
Of course the router needs to be connected to a broadband socket.
This diagram shows the principles of all powerline adapters. They effectively turn your mains wiring into one big Ethernet LAN.