Power Arduino and pass through relay

I currently have an Arduino Uno controlling a relay. The relay is connected to an extension-chord (to power on and off lamps). At the moment the Arduino is powered by battery.

Is there any way to power the Arduino and pass through current through the relay?

Probably yes, but need to see a drawing of your setup and more info on the relay, (coil voltage and current) and a part nr or link. HowToPost

Question doesn't make much sense (to me).

Relay, or relay module. You might be able to power Uno and relay module with a 5volt cellphone charger, connected to the USB socket. Or with a 7.5-9volt regulated supply, connected to the dc socket. Post images/links/code if you need help. Read the forum rules first. Leo..