Hi s_e_l_,
We recommend that you take a look at the article below since you're missing data after restoring your device from a backup.
About missing information after you restore your iPhone, iPad, or iPod touch with iCloud Backup - Apple Support
* Make sure that you correctly restored your iOS device from iCloud Backup. When you sign in to iCloud, it's not the same as when you restore from iCloud Backup. Learn how to restore from iCloud Backup.
* Make sure that iCloud Backup includes the kind of information that you’re missing. If you restore an iPad from an iPhone backup or vice versa, some data won’t restore.
* If an app asks for your Apple ID password during the restore process, make sure that you enter it. Some apps and services, like the App Store, need your password before they can restore purchased content. If you purchased content on multiple Apple IDs, enter the password for each Apple ID when prompted.
* If the restore process takes a long time to finish, or if you receive an error message, learn what to do.
* You can also check to see if you have any backups on your computer that may contain the data that’s missing.
* Sometimes the restore process may look complete, but your device is still downloading information in the background. To check this, tap Settings > [your name] > iCloud > iCloud Backup. If your device is still being restored, you see “This [device] is currently being restored and will automatically back up when it is done.”
As for purchasing more iCloud storage, you can learn how to do that here: Upgrade to iCloud+ - Apple Support
How to upgrade on your iPhone or iPad
1. Go to Settings, then tap your name.
2. Tap iCloud, then tap Manage Account Storage or Manage Storage. Or scroll down and tap Upgrade to iCloud+.
3. Tap Buy More Storage or Change Storage Plan.
4. Choose a plan and follow the onscreen instructions.
Lastly, if you'd like to start over with getting your data onto your new device, you'll need to follow the steps in the links below.
How to factory reset your iPhone, iPad, or iPod touch - Apple Support
1) Back up your device
Make sure to back up your device so that you can restore your data later.
2) Tap Settings > General > Transfer or Reset [Device]
Tap Erase All Content and Settings.
3) If asked, enter your passcode or Apple ID password
Forgot your passcode or Apple ID password?
4) Confirm that you want to erase your device
If you have an eSIM, you can choose to erase your eSIM or keep it. If you erase your eSIM, you need to contact your carrier to reactivate your cellular plan.
5) Wait for your device to erase
It might take a few minutes to completely erase your data.
Restore your iPhone, iPad, or iPod touch from a backup - Apple Support
1. Turn on your device. You should see a Hello Screen. If you already set up your device, you need to erase all of its content before you can use these steps to restore from your backup.
2. Follow the onscreen setup steps until you reach the Transfer Your Apps & Data screen, then tap Restore from iCloud Backup.
3. Sign in to iCloud with your Apple ID.
4. Choose a backup. Look at the date and size of each and pick the most relevant. After you choose a backup, the transfer starts. If a message says that a newer version of software is required, follow the onscreen steps to update. If you don't see onscreen steps to help you update, learn what to do.
5. When asked, sign in with your Apple ID to restore your apps and purchases. If you've purchased iTunes or App Store content using multiple Apple IDs, you'll be asked to sign in to each. If you can't remember your password, you can tap Skip this Step and sign in later. You won't be able to use the apps until you sign in with your Apple ID.
6. Stay connected to Wi-Fi and wait for a progress bar to appear and complete. Depending on the size of the backup and the network speed, the progress bar might need a few minutes to an hour to complete. If you disconnect from Wi-Fi too soon, the progress will pause until you reconnect.
7. Now you can finish setup and enjoy your device. Content like your apps, photos, music, and other information will continue to restore in the background for the next several hours or days, depending on the amount of information there is. Try to connect often to Wi-Fi and power to let the process complete.
Get help with restoring from your iCloud backup.
Please let us know the outcome after giving the above steps a try.
Thanks for using the Apple Support Communities.
Cheers!
