If you switch to HTML::TokeParser::Simple, I think you'll be happy with how much clearer the logic is.

use strict;
use HTML::TokeParser::Simple;
use LWP::Simple;
use URI;

my $url = 'http://www.reuters.com/newsEarlierArticles.jhtml?type=busin
+essNews';

my $stream = HTML::TokeParser::Simple->new(\get($url))
  || die "Couldn't read $url: $!";

while(my $token = $stream->get_token) {
    next unless 
        $token->is_start_tag('td') 
            and 
        ($token->return_attr('class') || '') eq 'earlyHeadline';

    my $next = $stream->get_token;
    if ($next->is_start_tag('a'))  {
        print URI->new_abs($next->return_attr('href'), $url), "\n";
    }
}
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