All of my tickets which do not have the status "New" do not show the "priority field" in the edit mode.

Why is this so?

When I created the issue the priority was urgent. After working a bit on this task the priority of this very task is now only high while other tasks need to be re-prioritized as well.

I added two Screenshots, showing one ticket where the priority is editable (status new) and another ticket where the priority is NOT editable (status in Bearbeitung).

Here is my environment:

Environment:
Redmine version 4.0.2.stable
Ruby version 2.3.3-p222 (2016-11-21) [x86_64-linux-gnu]
Rails version 5.2.2
Environment production
Database adapter Mysql2
Mailer queue ActiveJob::QueueAdapters::AsyncAdapter
Mailer delivery smtp
SCM:
Filesystem

Redmine plugins:
no plugin installed