Hello jglub,
Thanks for posting in the Apple Support Communities.
It sounds like you're looking to prevent notifications while you're using Do Not Disturb, correct?
If we have that right, let's check the settings enabled for notifications to appear while using a Focus status. You may have certain apps or contacts enabled to allow notifications, so check out the steps below to see if you can adjust this to your liking. You can also learn more details here: Allow or silence notifications for a Focus on iPhone - Apple Support
Allow or silence notifications from specific apps during a Focus
1. Go to Settings
> Focus, then choose a Focus—for example, Do Not Disturb, Personal, Sleep, or Work.
2. Tap Apps (or tap Choose Apps), then do one of the following:
• Allow specific apps: Tap Allow Notifications From, tap
, then select apps.
• Silence specific apps: Tap Silence Notifications From, tap
, then select apps.
You can also turn on Time Sensitive Notifications, which allows all apps to send time-sensitive notifications immediately.
We hope that helps. Let us know if you need additional assistance, and confirm which settings you have enabled already so we have a clearer understanding. We'll use any additional details to provide a more helpful response.
Take care!