I have heard people use this phrase. I have also heard people say "close the light". I dislike both phrases because I don't think they make sense and I feel they are just lazy English. For "close the light" I can grant that this might have meant something back in the old days when you might have had a lantern with an actual closing mechanism to shield the light.
Cut the power on doesn't make any sense to me. You can certainly Cut the power off, by literally cutting the wire that supplies the electricity. Cutting the power ON just doesn't seem to be a valid phrase. It's just people putting words together that sounds like it might mean something. To me it's like saying "unbreak the seal" and wanting people to understand that it means put the lid back on the bottle.
The proper phrase of course is "Turn the Power on" on "Turn on the Lights". "Cut on the Power" just comes across as being illiterate to me, or at least not understanding how electricity works.