Definitely not a good idea to buy a refurbished motherboard. The motherboard is a delicate and fragile and most important component. It is best to buy a new one. As for a backplate its a cover that you would find behind the computers that covers and displays which each ports are used for, like vga ports, usbs, rj45 port, audio ports and so on.
Buy a new motherboard, DO NOT buy a refurbished one please.
Hope that helps
after all, spending 40 $ extra is totally worth it because of warranty, safety and basically everything else.