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(1) Is there any way to fill up this empty space with nice logo's etc? See red arrow.
(2) Can the nr of messages button be moved to another place on the screen (to maybe fill up the space)
Out of the box you can’t modify the portal inn this manner. Customization of the portal is somewhat limited for sure. Here is an article you might find useful - what-can-i-customize-about-the-help-center
You forgot to include your screenshots :)
Sorry here they are :-)
Atlassian JSM Portal only allows basic modifications: color, message etc. If you want more customization you will have to use apps.
Here is one that could help you
Refined for Jira Cloud
My team had to build something custom and use the API to integrate the data, unfortunately, there isn't much you can do at all by default.