Howard Miller

No idea - it depends what you mean by "customisation". In the best case, they will have implemented the custom code as proper Moodle plugins and will not have changed any core code. Then it's a matter of making a thorough test of the upgrade (on a copy of the live site) with your custom plugins.

However, if they have modified core code then you have a big problem. You have found out why customising core code is a VERY bad idea.

The answer to my above question will probably define your way forward.

Most of the major problems I have had have been around trashing the server software on which Moodle runs rather than Moodle itself.

I've given up having any expectation of upgrading MySQL "in place", for instance.

AFAIK... the default location for backuptempdir is (moodledata)/temp/backup. Does that exist? What are its permissions. 

You can actually set the directory in your config.php by setting $CFG->backuptempdir = '/path/to/blah/blah';

...it may be worth setting that to somewhere you know for a fact has write access for your PHP/web server.

I assume we're talking about optional plugin...

https://moodle.org/plugins/gradereport_quizanalytics

Firstly, there isn't an advertised version for Moodle 4.4, so it's possible it doesn't work (properly). And... https://github.com/dualcube/moodle-gradereport_quizanalytics/issues/39

This might help...

http://www.syndrega.ch/blog/#php-and-dbms-compatibility-of-major-moodle-releases

 

The Devil is in the detail and depends on what PHP and MySQL versions you are running (and want to run). You will definitely need an intermediate step.