#!/usr/bin/perl
use warnings;
use strict;

use XML::LibXML;

my $html = 'XML::LibXML'->load_html(location => 'file.html',
                                    recover  => 1);
my $nav2 = $html->find('(//nav)[2]')->[0];
$nav2->parentNode->removeChild($nav2);
print $html->toString;  # c="d" is gone.
[download]

Or, more succinctly in XML::XSH2:

open :r :F html file.html ;
rm (//nav)[2] ;
ls / ;
[download]