If you have found this page looking for a solution for the Event ID 31 Windows Update failed to download an update error log then you came to the right place.
The probable cause of this error lies (at least in our case) in the missing mime-type on the IIS server.
To fix this issue, log in to your WSUS server an open IIS administration console.
From the left pane expand the Sites link and click on WSUS Administration. In the right pane double-click on the mime-types icon as you can see from the image below.
From the right Actions pane click on Add to add the new mime-type.
In the File Name extension field type .msu and in the mime-type field type application/octet-stream
Click OK and restart WSUS client for the changes to take effect. You should not receive Event ID 31 Windows Update failed to download an update error logs anymore.
Hope this helps.