use strict;
use warnings;
use XML::Twig;

my $xml = q(
<sites>
<site siteid="ONE">
   <name>name1</name>
   <address>address1</address>
   <contact>contact1</contact>
</site>
<site siteid="TWO">
   <name>name2</name>
   <address>address2</address>
   <contact>contact2</contact>
</site>
</sites>
);

my %handlers = ( site => sub { 
                                # my ($twig, $cnt) = @_;   
                                # $cnt is unuseful: iirc $_ is mapped $_[1]

				$_->delete if $_->children ('name[string() =~ /name2/]');

} );          
my $twig= new XML::Twig( PrettyPrint   => 'indented', twig_handlers => \%handlers);

$twig->parse($xml);
$twig->print;

# output:

<sites>
  <site siteid="ONE">
    <name>name1</name>
    <address>address1</address>
    <contact>contact1</contact>
  </site>
</sites>