The update problem is still present.
I tried now several times to clear all caches inside Joomla i found and I also rebuild the Update-Sites – but nothing – I still get the error Message

Warning
Update: Could not open update site #1 "Joomla! Core", URL: https://update.joomla.org/core/list.xml
Update: Could not open update site #3 "Accredited Joomla! Translations", URL: https://update.joomla.org/language/translationlist_3.xml
Update: Could not open update site #4 "Joomla! Update Component Update Site", URL: https://update.joomla.org/core/extensions/com_joomlaupdate.xml
Update: Could not open update site #59 "RocketTheme Update Directory", URL: http://updates.rockettheme.com/joomla/updates.xml
Update: Could not open update site #60 "System - Helix3 Framework", URL: http://www.joomshaper.com/updates/plg-system-helix3.xml
Update: Could not open update site #61 "Helix3 - Ajax", URL: http://www.joomshaper.com/updates/plg-ajax-helix3.xml
Update: Could not open update site #62 "Unite Slider", URL: http://unitecms.net/update/unite-slider.xml
Update: Could not open update site #63 "Quix Update Site", URL: https://www.themexpert.com/index.php?option=com_digicom&task=responses&source=release&format=xml&provider=joomla&pid=116/extension.xml
Update: Could not open update site #64 "FOF 3.x", URL: http://cdn.akeebabackup.com/updates/fof3.xml

Very curious.

I also checked if I could update to Joomla 3.8.5 manually, but on the actual webspace I only allowed to upload files up to 8M and the recommended Upgrade Package from Joomla is bigger than 11M.