It looks like the syntax of the HTTP::Request::Common's POST function, except you replaced the call to POST with the string POST. And then you commented out that line entirely! The following is closer to what you want

use HTTP::Request::Common qw( POST );

my $request = POST($url,
        action => 'Count',
          name => 'datafile',
  Content_Type => 'multipart/form-data',
       Content => [ 'file' => [ $ballot_file ]]
);
[download]

But you're not quite there yet. You have content fields outside of Content. Fixed:

use HTTP::Request::Common qw( POST );

my $request = POST($url,
    Content_Type => 'multipart/form-data',
    Content      => [
        file   => [ $ballot_file ],
        action => 'Count',
        name   => 'datafile',
    ],
);
[download]

The example in the docs is

  POST 'http://www.perl.org/survey.cgi',
       Content_Type => 'form-data',
       Content      => [ name  => 'Gisle Aas',
                         email => 'gisle@aas.no',
                         gender => 'M',
                         born   => '1964',
                         init   => ["$ENV{HOME}/.profile"],
                       ]
[download]

I don't know how you got from that to what you posted. Or maybe you didn't bother to read the docs...