Empty nodes are being created from constructs like $node->{children}->{data}->{type} creates the intermediate data structures in that chain, and therefore an empty node. You can fix this by factoring out those lookups:

my $children = $node->{children};
my $child_type = undef;
if ($node->{children} and $node->{children}->{data}) {
  $child_type = $node->{children}->{data}->{type};
}
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then just use $child_type.

In your attempts to fix the code you have both a dclone and a freeze. The freeze will result in a string that cannot be accessed properly which will break the rest of the code. The dclone itself is enough. But note that dclone has a considerable performance penalty. So fixing the bug that causes the blank nodes is better than using it.

On a general note, I reiterate that Tree::Nary is a bad idea. What you are basically doing is trying to code Perl as if it was C. But Perl is not. C structs are efficient and have essentially no overhead. In Perl an anonymous hash ref takes 92 bytes. Before you put anything in it. (You can use Devel::Size to get a sense for this.) Complex structures built out of anonymous hashes come with a considerable memory and performance overhead. Furthermore virtually anything you can do with them can be done with far simpler native Perl structures, in much less code.

(Speaking of less code, your style is rather..verbose. I'd highly recommend finding ways to make your comments shorter and your code more descriptive. Code Complete has some excellent advice to help you with that.)