See WWW:::Mechanize and credentials and the WWW::Mechanize::Cookbook especially Fetch a password-protected page.

Then try do fetch the page manually. Double check if Firefox has already stored a username and password for you.

Next step is to build your code in small steps to see whats going on:

#!/usr/bin/perl
use strict;
use warnings;
use WWW::Mechanize;

my $mech =  WWW::Mechanize->new( stack_depth => 1,
                                 timeout => 180,
                                 autocheck => 1 );



# Part 1
my $url = 'http://somewhere.tld';
$mech->get($url);

__END__

# Part 2
#
my $username = '';
my $password = '';


$mech->form_name('loginform');
$mech->field(login => $username);
$mech->field(passwd => $password);
$mech->click(); # Which button?

# Part 3

my $outpage = $mech->content();
my $outfile = 'output.html';
    # Always use the three argument form of open
open(OUTFILE, ">", $outfile) or die $!;
print OUTFILE "$outpage"; 
close(OUTFILE) or die $!;
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BTW: Please read Markup in the Monastery.

Hope this helps.