The ability to publish, save, and edit content in App Manager is dependent on:
- The user's permissions (Contribute, Contribute + Approval, or Full Control)
- The list's settings:
Just because a user has Approval permissions does not mean they have Contribute permissions (add new or edit items). Users with just Approval permissions can edit the item to only change the "Approval" status but nothing more.
Users with Approval permissions in a SharePoint security group, still need to be in another SharePoint Security group who can contribute to SAVE and PUBLISH in the AppManager.