With the courtesy of Jim, the author of Peach OSI, I have written this small guide for fixing some glitches in the new software center in 16.04(It seems to like to remove entries for software that wasn't installed by the new software center - this is a problem from Ubuntu and not Peach OSI), so if you can't wait for an updated version of Peach OSI 16.04 with the problems solved, then try the following solution - it worked for me:
Before we begin, make sure you have an internet connection.
Open the old software center and search for and install "Debian Package Search" - don't use " ".
Run it(it might start by itself after installing). Give it a minute and then it should have a pop up window appear asking if you want to run the app - cache repair(that is what we want to do), followed by a terminal window.
You can close the old software center while this goes on.
When its finished click on Close and reboot your machine the normal way.
When your system is restarted, you need to open Software Updater from the dock in the bottom of the screen, and do the partial upgrade. Let it run - it may take an hour or more depending on your internet connection.
It'll ask questions to remove older deprecated files - let it remove them.
Reboot and run update again until you get the report from Software Updater, that the system is updated.
Then run the new "Software" center and see if there are any updates there - which there should be.
After you are sure you've completed all the updates reboot again and boot to the new Linux Kernel 93 and then run Ubuntu Tweak from settings, to remove all the cruft and old kernels.
Then run Grub Customizer and adjust Grub to how you want it to look and then reboot again checking your handy work.
You should now have a good clean install and the Debian Package Search program should always notify you if your app cache gets targeted by the new Software Center app again and will automatically notify you if there is ever an issue.
Hopefully Ubuntu will address this issue with their new software center, sooner than later.
If the update center still will not work we may need to update the xapian cache.
Open a terminal and type
Let it run - reboot - try to update again.