That pergola was built a few years ago by Jason Roberson of Charlottesville, VA. I'd write to him and ask how it is holding up and what he might do differently.
One of the photos shows that the top joint was pinned laterally. See:
Note that the exifdata in most of the images date the photos as 2013, and don't show the lateral pin. But the photo I cropped and show above is from 2015 per exif data.
The slots in the bottom of the posts look like mortises for 1/2" steel plates sunk into a cement footing.
You can see high res versions of his photos by starting the slide show, then right-click -> View Image (firefox), then edit the image url to delete the questionmark and subsequent characters. For example:
Looks a little like a bullet connector / terminal.
If you have access to the other end of the connector that you think is broken, you could disconnect it & test the wire to see if it's really broken using a multimeter (set to ohms / resistance), or test it in place with the power off. Maybe that wire's not the problem, there are a lot of other electronics in there too.
Most light fixtures just use cheap wire nuts to twist two wires together, you could just cut & strip both ends of the wire and use a wire nut.
PS Don't forget to check why it was pinched & don't do it again.
Best Answer
That is the drain point for either the heating system ie radiators as it is the lowest point or it could be the drain for the boiler.
What is on the other side of the wall - that may help, but I would go with the drain for the radiators.