HoMM3 HD bug reports.

For some time now, the HD mod has stopped updating; what should I do?

Judging by the message, HD is trying to update from the wrong location.
You can try placing this file: https://www.dropbox.com/s/t9vzs83iwmx92fh/%23common.ini?dl=0
in the _HD3_Data\Settings folder.
This file contains the addresses of the update servers. Although it's strange that it "corrupted" for you (and, especially, that it didn't fix itself after the update). Perhaps the download is being blocked by a firewall, or the launcher doesn't have the rights to write files in this folder.