I have an extension of my pergola into my backyard that is supported by a single pole which is now showing signs of rot and damage.
After having a close look at your images, it's my recommendation that this structure is pulled down and, if you wish, can be constructed again with new timber. I do not see any timber in your images that's structural integrity hasn't been compromised by rot and could be saved. Wood hardener is for aesthetical repairs only, and if you were to cover up this damage with putty and paint, the whole thing could collapse at any stage onto an unsuspecting victim. Not only is the post rotted out almost completely, but the beam has an enormous amount of rot and looks ready to collapse under its own weight.
I'm all for saving money where you can, and I'd be more than happy to assist if I thought this could be fixed, but it can't, and you're best to re-build it if needed.
Looks almost like our pergola when we purchased our house.
As far as that post - I've used a carjack with wood/steel to prop up the horizontal beam and remove a post.
But even then it looks like compressed rot on the horizontal member requiring some galvanised plate prior to the post being attached.
But noticing Mitchell's post above, I would concur with his diagnosis - not for resuscitation.