That is correct.In the box with the blanking plate i have separated the 2 black wires.
This was done when i plugged a lamp into the double socket.
The lamp still works.
So with these 2 black wires separated & the lamp still working tells us that these 2 sockets are not connected to each other.
I assume that is correct ?
Probably not. It appears to be a spur so no more sockets are allowed (unless you fit a Fused Connection Unit before the existing socket).So now to put a new double socket in this room can i join my new length of wire "5 meters" into the same double socket connections.
This then means leaving the item with the blanking plate untouched.
Is there a socket on the other side of the wall to which it might be connected?
You could just have a plug on your new 5m. cable and plug it into the socket.
The blank plate might be able to have a socket fitted instead plus another cable to a new socket, BUT you will need to get an electrician to determine what the situation is and if it will be safe.