So, as has been suggested by other posts that have this “Socket error on client” error, my problem was authentication.
I could not get the logins section of MQTT to function at all. My garadget would not use the username and password I had added there. Or rather, when using that user and pass, MQTT would not authenticate them.
I also couldnt get it to use users I’d added to HA. I did however get it working using the admin login of HA. Not ideal, but its working for now.