This should be a good start; it adds a 3rd site with an id attribute and name element. You should try to add the address/contact elements. See also the XML::Twig tutorial.

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 $twig = XML::Twig->new();
$twig->parse($xml);

my $root=$twig->root;
   
my $site = $root->first_child('site');
my $cp = $site->copy();
$cp->set_att('siteid', 'THREE');
$cp->first_child('name')->set_text('name3');
$cp->paste('last_child', $root);

my @sites = $root->children;

for my $site (@sites){
    print "Site ", $site->att("siteid"), "\n";
    print "name ", $site->first_child('name')->text(), "\n";
}

__END__

Site ONE
name name1
Site TWO
name name2
Site THREE
name name3
[download]

Updated to use the copy method.