This is very long winded and wordy, but I am trying to cover as much as I can in this response. If there is anything that you do not understand or if you are uncomfortable with trying any of this, please post back and I will answer your questions as best I can. There are many very knowledgeable people here that will hopefully offer help as well.
I backup with iCloud and with iTunes. I also sync my devices with iTunes because I have photos on my Mac that I want to sync and I have documents that are stored on the devices that I have moved back and forth with file sharing since not all apps work with iCloud. I also have most of my music collection on the iPad that was ripped from CD's and I don't use iTunes Match so I have to sync those songs to the device.
This is the approach that I would take and be forewarned that it can be very, very tedious and it will take some time. It could take days to sort it all out.
The first thing that I would do is backup with iCloud and then erase the device in Settings>General>Reset>Erase all content and settings. You will have to go through the setup process again. You will be given the option to restore the iCloud backup. Go ahead and say yes to that option. All of your content will be downloaded from the cloud and that can take hours if you have a lot of content on the iPad. This will not restore any synced content that you have on the device, so you will still have to sync photos, music, books or any other non iTunes purchased content back onto the device. If you do not sync with iTunes, this step will be a non issue for you so you will not have to worry about it. However, if you do sync some content with iTunes, at this point, I would forgo that step for now. Use the iPad and see how it goes. If it all goes well, then sync whatever you were syncing back to the device. If the problem appears after syncing with iTunes, assume that there is something corrupt in iTunes.
If all of that fails to do any good, you can backup again, erase the device again, go through the whole set up process but do not restore the backup. See how that goes. If everything goes OK, sync with iTunes - but only if you sync content with iTunes - and see how that goes. If everything works out OK, you can assume something in the backup was corrupt and causing the problem.
If you use iTunes only, always make sure that you backup the iPad one last time before you do the clean install from iTunes. Make sure that you transfer all purchases and apps into iTunes as well. Then restore as new and use the device for a while and see how it goes. Slowly add a few of your purchased apps at a time. See how it goes, add more apps, see how it goes. When you get to a point where you believe that things are good to go, try restoring the backup which will restore all of your app data and the device settings. If the problem comes back, it has to be something in one of the apps or a corrupt setting on the iPad. Restore as new again, and download or sync your purchases back from iTunes and start all over again. Do not restore the backup.
What this amounts to is, doing things by process of elimination. Like I said, it could take days to sort it all out. I am a glutton for punishment and I don't mind going through all of the tedious rigamarole involved.
Raoul_Duke wrote:
How would one move the data from the "corrupted" back up?
Unfortunately, you may never know what item, data, file, photo, song, etc. is corrupt and you cannot move selective data back from the backup. It is all or nothing. If you feel that the backup is corrupt or if you go through all of the steps and know that the backup is corrupt, then you cannot use it at all and you have to start over again.
This is one of the reasons why I keep copies of my most important files on my Mac, and not just in the iPad backups. I can selectively add files back to the iPad if I need to start all over again.
Sone links that might prove to be helpful.
iOS: Understanding 'Erase All Content and Settings' - Apple Support
How to erase your iOS device and then set it up as a new device or restore it from backups - Apple Support
How to back up your devices using iCloud or iTunes - Apple Support
Use iTunes to restore your iPhone, iPad, or iPod to factory settings - Apple Support
Re-download or transfer your iTunes Store purchases from an iPhone, iPad or iPod to a computer - Apple Support
Download past purchases - Apple Support