Unable to connect to patching server.

The process will just be called Client.exe, but it shouldn't be running prior to you starting it. The patcher does just tend to start pretty fast as it's fairly light weight.

I would suggest you try disabling any firewalls/antivirus or other security programs you might be running and see if that fixes the problem - if it does you can re-enable them and try to figure out which one is blocking the patcher.

As far as I'm aware there shouldn't be any way for a patch to cause this directly. You could try download a new copy of the installer and reinstall the game (back up your Content.ggpk file first in the games install dir, so you don't have to re-patch everything).

You could also try using an alternate patching port, in case something upstream of your computer has blocked the port somehow. To try this, create a shortcut to the Client.exe file (or use the default shortcut on your desktop), right click the shortcut and go "Properties", then go to the "Shortcut" tab, and append " --patching-server pathofexile.com:3724" to the end of the "Target" box. So assuming a default install dir the "Target" box will look like:

"C:\Program Files\Grinding Gear Games\Path of Exile\Client.exe" --patching-server pathofexile.com:3724

(or possibly with "Program Files (x86)" if you have 64 bit OS). Then run the shortcut normally.