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I'm not really sure whether notes are backed up or not.
At one time it was thought none of the iCloud sync service data (contacts, calendars etc) are not backed up, however a while back some reports suggested they might be, but were being overwritten after being restored, by iCloud syncing. At that time I suggested a solution that might help to recover contacts (see below) and one for calendars if this were the case.
It now appears that contacts do back up since others have reported my suggestion has enabled them to recover lost contacts, there is also some evidence to suggest the same might be true for calendars, but I have no information about notes.
You might try adapting the following suggestion for contacts and let us know how it goes. I can re-write it for notes later for you if you aren't sure what to change, but I need to do something else right now.
The difficulty in restoring iCloud data from a back up is that the back up is older than the current version on iCloud and anything that is restored gets removed again when an iCloud sync occurs.
- Ensure the device is connected to your local network.
- Start the restore from iCloud back up.
- Allow the devices settings to restore (watch for the confirmation of completion).
- As the apps begin to restore pull the connection to your router WAN port (the one that goes to the telephone point).
- Wait for the time out confirmation.
- Navigate to settings > iCloud and turn off contact syncing (keep contacts when prompted).
- Reconnect the router to the internet and let the restore process complete.
- Delete any iCloud contacts from iCloud.comon a computer.
- Navigate to settings > iCloud and turn on contact syncing (merge contacts when prompted).
May I suggest that you carry out a back up to iTunes before you try anything, that way should things become even worse, you can at least restore back to where you are now.