Failure to download extra data files: flashplugin-installer

I had the same problem and have fixed it with the following two steps from the command line:

1. Removing flashplugin-installer completely with the following command

   sudo apt-get remove --purge flashplugin-installer
   

2. Re-installing flashplugin-installer again with the command

   sudo apt-get install flashplugin-installer
   

Another command to force the download to be reattempted:

sudo pkexec /usr/lib/update-notifier/package-data-downloader

I just got this to work in 16.04. This way you don't have to uninstall/reinstall the package. It's the same thing that would happen if you click the "run this action now" button in the GUI.

Running sudo pkexec /usr/lib/update-notifier/package-data-downloader I received the following error (links obfuscated due to ask policy):

flashplugin-installer: processing...
flashplugin-installer: downloading _http://archive.canonical.com/pool/partner/a/adobe-flashplugin/adobe-flashplugin_20161213.1.orig.tar.gz
Err:1 _http://archive.canonical.com/pool/partner/a/adobe-flashplugin/adobe-flashplugin_20161213.1.orig.tar.gz
  Could not open file /var/lib/update-notifier/package-data-downloads/partial/adobe-flashplugin_20161213.1.orig.tar.gz - open (2: No such file or directory) [IP: 91.189.92.191 80]
E: Failed to fetch _http://archive.canonical.com/pool/partner/a/adobe-flashplugin/adobe-flashplugin_20161213.1.orig.tar.gz  Could not open file /var/lib/update-notifier/package-data-downloads/partial/adobe-flashplugin_20161213.1.orig.tar.gz - open (2: No such file or directory) [IP: 91.189.92.191 80]

E: Download Failed

I manually downloaded the file from the URL specified, created a new 'partial' directory in /var/lib/update-notifier/package-data-downloads, moved the download to there and then ran the sudo pkexec /usr/lib/update-notifier/package-data-downloader command again. Which I think appeared to work (Ubuntu 16.04)