What is the purpose of the PRODUCTIVITY App?

I suspect that you mean the Productivity folder.

Also see: Organize your apps in folders on iPhone - Apple Support

It is just a folder of similar apps. If it is on your Home Screen you can add any app that you want to that folder.

You can also drag apps out of the folder to the Home Screen.

If it is in the App Library, go to the App Library (swipe left past your last Home Screen). From there you can touch and hold an app and slide it to the Home Screen.

It's just a way to organise the apps, you can create folders, give them a name and add apps in it. The iphone comes with some pre-made folders that have a name like productivity or utilities usually containing office or work apps. You can create your own and free the space on the screen moving the apps into the folders

Organize your apps in folders on iPhone – Apple Support