#! /usr/bin/perl
use strict;
use feature qw{ say };
use warnings;

use XML::LibXML;

print "Please specify node c content: ";
chomp( my $new_text = <STDIN> );
my $file = 'xx.xml';
my $dom = 'XML::LibXML'->load_xml(location => $file);
for my $node($dom->findnodes('(/header/id/c)')) {
    $node->removeChildNodes;
    $node->appendText($new_text);
}
#print $dom;
open (my $Output, "> $file") or die "Could not write $file";
print $Output $dom->toString();
close ($Output) or die "Could not close generated $file ";
[download]

<?xml version="1.0"?>
<header>
   <id x_id="1">
      <a/>
      <b/>
      <c>as</c>
   </id>
</header>
[download]

<?xml version="1.0"?>
<header>
   <id x_id="1">
      <a></a>
      <b></c>
      <c>something</c>
   </id>
</header>
[download]

The difference between your observed and expected output appears to be down to canonicalisation. See eg:

#!/usr/bin/env perl 
use strict;
use warnings;
use XML::LibXML;


my $dom = XML::LibXML->load_xml (string => <<'EOT');
<?xml version="1.0"?>
<header>
   <id x_id="1">
      <a/>
      <b/>
      <c>as</c>
   </id>
</header>
EOT

print "\nNot canonical:\n\n" . $dom->toString;
print "\nWith canonical:\n\n" . $dom->toStringC14N;
[download]