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Dear RoMaWien,
Good day! Thank you for posting in Microsoft Community.
From your description, it seems that you have set target audience on SharePoint navigation via browser, however, it didn’t work on the SharePoint on mobile device. And you have tested with different devices and same problem still persists.
If my understanding is right, for the situation you encountered, we do understand the inconvenience caused and apologize for it.
We have also tested it on our side, we went to SharePoint Online to set target Audience, then we went to SharePoint for Android, we used the user account who don’t such permission, we still cannot find the quick links. We cannot reproduce the same situation as yours.
Given this situation, we do understand the inconvenience caused by it. Since we cannot reproduce the same situation as yours and two different devices have the same problem, it may need to be checked from background. The best way to troubleshoot the issue is to report the issue to the related Team. We’d suggest you contact your office 365 administrator to go to Office 365 Admin Center-> Support-> New Service Request, the support engineers there have the correct escalation channel and this is the most efficient way to report such issue. You can refer to this article to check the way how to raise a ticket. Ways to contact support for business products - Admin help. The engineers in the related team may help recover it from background.
If the scenario above is not consistent with yours, you can also post back and point that.
Thanks for your cooperation.
Sincerely,
Sukie | Microsoft Community Moderator