To get the debug to show everything do the following:

use LWP::Debug qw(+);
[download]

I have tidied up the code a bit... and changed your proxy line (by adding 'http://' to the proxy address). This now works from behind my proxy server and I guess is the solution to your problem.

#!/usr/bin/perl

use strict;
use warnings;
use Data::Dumper;
use LWP::UserAgent;
use LWP::Debug qw(+);
use WWW::Mechanize;


my $total_count = 0;

my @keywords = ('simpsons',
'diehard',
'fantastic+four',
);

foreach my $keywords(@keywords){
  my $url ='http://news.google.co.uk/search?q=~%22'.$keywords.'%22&num
+=100&hl=en&safe=off&start=0&as_qdr=all&filter=0';
  my $mech = WWW::Mechanize->new();
  $mech->proxy('http','http://127.0.0.1:8088');
  $mech->get($url);
  print $mech->uri."\n";
  my @links_to_check =  grep { $_->url() !~ /google/i}   $mech->find_a
+ll_links( url_regex => qr/\./i );

  foreach my $links_to_check (@links_to_check){
    $total_count++;
    print "$links_to_check \n";
  }
}
print " $total_count news items found \n";
[download]

When you print $links_to_check you will get a response like WWW::Mechanize::Link=ARRAY(0x2a01214), this is because you are printing a Mechanize Link object (see WWW::Mechanize::Link), you can dig down a bit to get the url etc and the documentation should help you with that.