Recovery of Photos

As that iTunes backup didn't contain your photos, and you never even had Photos turned on for an iCloud backup, then I'm sorry to tell you but those photos are gone.

For future reference, see the following articles regarding iCloud storage:

Buy more iCloud storage

Set up iCloud on your iOS devices or your Mac or Windows computer

Set up and use iCloud Photos

Manage your iCloud storage

One last point, always backup your photos etc. to your computer, not just iTunes or iCloud, and then back your computer up to an external hard drive or Flash drive.

Do you backup your photos to iCloud? If so then check there.

Just to clarify, when you say you "restored" your phone, do you mean you restored it to factory settings, or you restored a backup to your phone?

You're welcome. I just wish the news could have been better. Remember, backup to your computer and an external drive, not just iTunes and/or iCloud.