Hi -I have BT broadband (FTTC). I have a master socket in hallway and a standard wi-fi router placed there. I have a office at the end of the the garden and there is a ethernet cable between house and outbuilding. The cable is not connected at either end. Annoyingly the cable in house does not reach to master socket and there is no other socket in the house.

I am wondering how to get wi-fi to office? I cant extend the ethrnet cable in house to reach the router without a lot of hassle. Can i get a mesh network with 3 routers and place one near master socket, one near where the cable ends in house and third one at the other end of cable in outbuilding? I don't know if the mesh network "outputs" internet so i can plug in the ethernet cable? do i need any other euqipment other than the mesh routers if this works? I dont want to use power line adapters as i didnt have a good experience in the past. 

Do i need a "modem" with the mesh to put near the master socket?

Thank you