I think that my PS4 isn't working correctly in standby mode

[JOKA_]

Gremdude said:

You are not alone. Download and uploads happen when you turn on the PS4. There is no background downloading, uploading OR trophy synching when it is in Standby. This is probably the reason we don't have the suspend/resume feature either.

In the settings under 'Automatic Downloads and Uploads' it says that if you have PS+ it will "download update files for applications" while in standby mode

Are games not applications?  Seems like some sketchy wording if thats the case