Same issue with KB4038777, KB4038803 (optional) and error code 80073701: I ran the Troubleshooter and it fixed: Service registration is missing or corrupt; Problems installing recent updates (listed twice). However, "Not Fixed" was Windows Update error 0x80070005(2017-10-06-T-09_01_32A).
I then ran and installed the Update Readiness Tool. It installed a Hotfix for Windows KB947821.
What should I do next? Try and install the Windows Update or another repair step? Thanks for your help.