Thank you for the response Christina!
From TECH41865:
To reiterate the Client failing to run on the Schedule; Example: Software Update Task/Policy is set to run at 3am tonight. However, they miss the run due to client needing a reboot, being turned off or unable to get the Policy/Package in time. The Default Software Update Policy schedule will be the next attempted run, and if that run process fails; the next run is ASAP regardless of any schedule defined by Patch Management.
This reads like if my patches fail to install Monday evening, then they will try on the weekend (my Default policy is set for weekends), and if that fails, the patch will attempt to install the next time the client checks in (ASAP). Which is what I want it to do, but you are saying it will only install on Monday evening. Have I misinterpreted TECH41865?