The root cause of this is most likely the Windows Fast Startup and Hibernate features. See here and here for information on how to disable these features in Windows. As background, these features turn a shutdown operation into a suspend-to-disk operation. This has the desired effect of reducing subsequent startup times. Unfortunately, the undesirable consequence is that it becomes unsafe to share partitions between OSes, since they'll be in an inconsistent state when the non-Windows OS boots; and then when Windows boots, it won't know how to handle any changes made by the non-Windows OS. Thus, file corruption can result. This can affect the EFI System Partition (ESP), where both Windows and Ubuntu boot loaders are stored. In your case, I'm speculating that these features were active (as they are by default), which resulted in what looks to Ubuntu like filesystem damage.
The solution, if I'm right, is to disable those features. You might need to reboot back into Windows for the change to take effect. Thereafter, you should be able to safely read and write the ESP.
If the partition still shows damage after disabling these features, there is a native Linux tool for repairing FAT filesystems, dosfsck. You can use that instead of CHKDSK in Windows, but you need to either pass it the -a option, as in sudo dosfsck /dev/sda1 to check /dev/sda1; or use more advanced options to approve individual changes. That said, if you're seeing filesystem damage, you might want to back up the partition first. Using a file-level backup tool (cp, tar, or zip, for instance) should be sufficient. A backup will provide you with a fallback position should a filesystem repair operation make matters worse. (This assumes that all the files are successfully read and backed up, of course!) In that case, you can create a fresh filesystem and restore all the files.
I've never tried using CHKDSK on an ESP in Windows. My suspicion is that it could be done as follows:
- Open an Administrator Command Prompt window.
- Type
mountvol E: /S to mount the ESP as E: in that window. (Change E: to whatever you like.)
- Type
CHKDSK E: to run the check.
As I said, I've not tried this, so I can't promise it would work. If I'm right about the cause of your problem, this procedure alone will be useless, even if there's filesystem damage; you'll still need to disable Fast Startup and Hibernate to make the system safe for use in a multi-boot environment.