Given that you have this code already (which would be displayed in your post as follows, if you just put <code>...</code> tags around the code):

my $mech = WWW::Mechanize->new();
$mech->get($url);
$mech->form_name('loginform');
$mech->field(login => $username);
$mech->field(passwd => $password);
$mech->click();
my $outpage = $mech->content();
open(OUTFILE, ">$outfile");
print OUTFILE "$outpage";
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If it's true that the contents of "$outpage" include a string that matches this regular expression:

/You have \d+ unread messages/
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Then you just need four more statements: match and capture that string in $outpage, open your "final" output file, write the captured string to the output file, and close the file. If you need to see the code for that...

if ( $outpage =~ /(You have \d+ unread messages)/ ) {
    open( FINAL, ">", "make_up_a_file_name" ) or die $!;
    print FINAL $1,"\n";
    close FINAL;
}
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If that regex doesn't match, you'll need to check the actual contents of "$outpage", and figure out how the regex would need to change in order to match the desired phrase.