Problems regarding the 80072EFE is most common associated with clients not able to communicate with the WSUS-server.
I have twice encountered this problem with my WSUS-server:
Windows Update fails: error 80072EFE
First time, this was because I failed to NAT port 80 to our WSUS-server, even tough I ‘m forcing all clients to use https on port 443, it seems that port 80 also needs to be open.
Second time was today, this time I had performed an upgrade of the WSUS-server and then rebooted it. After this I started getting this error again on all of our clients, but now everything seemed to work, both the HTTP and HTTPS ports seemed to respond, but after taking a closer look in IIS, I discovered that during the update/reboot that the SSL certificate for the WSUS site was not selected anymore.
After selecting the certificate and do a restart of the services, everything worked as usual again.