HI@christoph2003, Welcome to the HP Support Community!
I understand that you are getting bad image quality while connecting TV via HDMI cable.
Do not worry. I will try to fix the issue.
Have you tried with a different HDMI cable?
Please refer to the below link to troubleshoot the issue.
https://support.hp.com/in-en/document/bph05236
Please update the graphics driver and BIOS.
Keep me posted. Happy to help!
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Have a great day!