iPhone Backup – How to Recover Files Without Restoring

It depends whether the remaining files you have are from an encrypted backup.

It won't be feasible to read your backup data if you have an encrypted backup with no Info.plist, Status.plist, or Manifest.mbdx files

However, if your backup wasn't encrypted, it will be possible to read that data. You'll find that whilst the filenames are unhelpful (they're SHA-1 hashes of the file's internal path), the file contents easily readable. It's only the filenames that'll be lost.

If you're looking for images, try renaming a bunch of the files to have .JPG extensions, and see whether Windows Explorer or macOS Finder previews them for you.

There are a few approaches for digging into what's there in bulk. You mentioned using iPhone Backup Extractor, and there's a guide on how a use "Bulk Rename Utility" to read backup data when in this situation.

Apple provides no interface to pull individual items from the backup. Whether through iTunes or iCloud, backups are intended to be restored as a whole, replacing the content on the device at the time of the restore.

Is it possible that the image is also in your Photo Stream? That's probably the best possible solution if you do not wish to go through the steps to restore the backup to a device (your device or another one) to retrieve the photos you are missing (and then restore things again from a more current backup if desired).

If you do need to restore the whole backup, erase the phone (Settings -> General -> Reset -> Erase All Content and Settings) then you can restore from iCloud. When restoring from iCloud, you can choose from multiple recent backups.