The way you should think about app authentication, is that when you are authenticated as an app, you can act like a regular user with access to only this app. As opposed to admin users, regular users cannot delete items, files and other content not created by themselves. They can only delete things they have created themselves.
For this reason, when using app authentication, you can only delete items you have created while authenticated as the app.
For the problem about not being allowed to add a file to an item, if you are sure the item is indeed in the app you are authenticated as, check that the box that allow users to add files is not unchecked in the advanced settings when editing the app. When authenticated as an app, you follow the same rules as regular users.