We are running Wordpress 5.2.1 with CiviCRM 5.12.4. In trying to upgrade to 5.13.5, the database runs through the updates, but then we get the message: "The site is experiencing technical difficulties. Please check your site admin email inbox for instructions." Afterwards, the bottom of the page says "Powered by CiviCRM 5.13.5", but under System Status we get the warning "CiviCRM 5.12 has reached its end of life." indicating the upgrade process didn't run through to the end. Looking at the error_log, there is a fatal PHP error and stack trace which looks like" "PHP Fatal error: Uncaught Error: Class 'CRM_Sparkpost' not found in /home/britstatt/test.sdcbcdream.org/wp-content/uploads/civicrm/ext/com.cividesk.email.sparkpost/settings/Sparkpost.setting.php:8, referer: https://test.sdcbcdream.org/wp-admin/admin.php?page=CiviCRM&q=civicrm%2Fupgrade%2Fqueue%2Frunner&reset=1&qrid=CRM_Upgrade"
We have been using the Sparkpost mail extension since summer of 2016 and have never experienced an issue during the many CiviCRM upgrades since then, so I don't think this is a particular issue with that extension, although I will also ping cividesk. Since the upgrade process has always worked seamlessly, I wanted to throw this out there to find out what might be the matter.
Fwiw, I originally encountered the issue when upgrading from 5.10.3 to 5.13.5, but didn't dive into the error logs. So I ran the upgrade from 5.10.3 to 5.12.4 successfully, thinking doing the upgrade in steps might help. That worked fine, but the subsequent attempt to upgrade 5.12.4 to 5.13.5 generated the error again and I finally looked into the error log.
Our live site is running fine at 5.10.3, but I always run upgrades on a test mirror site, so this is not critical except we really want to move to 5.13.x which is supposed to be the next ESR, I thought.