- Install a single-gang new-work box1 in the rectangular hole to the left.
- Run a wire behind the wall to the new hole on the right.
- Make the junction on the left
- Wire up the new fixture on the appropriate box in the hole on the right
- Put a blank cover plate2 on the box on the left
- The fixture itself should act as the cover for the hole on
1Use a new-work box similar to this:
Image courtesy of Lowes.com, no endorsement of vendor or brand intended or implied
Because you can nail straight through that flange on the left into the stud in the hole on the left. Any sort of flange-mounted box will work just fine, this is simply an example. You do not want a new-work box that has 2 nails through the side - there won't be room to swing a hammer at them.
2Use a blanking plate like this:
Image from Lowes.com, disclaimer, disclaimer, disclaimer
I chose a metal one because it was more visible in the image, but you can use any color or material you choose. You can even paint/wallpaper the cover plate to match the rest of the wall. I picked a duplex cover plate since if you use the right-hand set of screws to hold it on to the box, it should cover both sides of the hole. You may, depending on the size of that rectangular hole, need to use a triple-gang blank cover plate.