Given you are using Excel data sources, your options will be limited. Basically, at run-time, the app will try to read the Excel file in any tabular object or function that references the source.
What you're experiencing is the fact that the sources are being read (probably by another screen's control), regardless of whether or not the user has access and the error is being thrown at that time.
I would explore the Errors function to see if this may help (can't say I have done this with Excel, however). https://docs.microsoft.com/en-us/powerapps/maker/canvas-apps/functions/function-errors
A few other alternatives:
- Refactor your app to use collections instead of direct data calls; this way you can explicitly collect from the specific Excel file at a specific time, instead of it trying to poll/read the source
- Consider a better data source; Excel is notoriously fragile and insecure - SharePoint is an upgrade, and CDS would be best, but there are implications for licensing and development
- Develop multiple apps with the specific data sources only; obviously not a viable option is the UX is identical in all apps (with different data), since changes would need to done X times per X apps, but may be a better idea if each data set group has specific things (like basic actions vs. admin actions, for example)
