Welcome to The Monastery...

You are dealing with %self keys where you actually needed to deal with the referenced anonymous hash $self in the constructor and the getClusterName() methods instead because that is what you intended by initializing an anonymous hash via "$self={}". Using strictures is note-worthy at discovering such mix-ups.

#"Cluster.pm"
use strict;
use warnings;
package Cluster;


sub new
{
  my $class = shift;  #The first argument passed to the
                      #new method is the name of the class
  my $self = {};

  $self->{_clusterName} = shift; #Read data passed to constructor
  $self->{_verbs} = [];

  bless $self, $class;
  return $self;
}

sub getClusterName
{
  my( $self ) = @_;
  return $self->{_clusterName};
}
1;
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use strict;
use warnings;
use Cluster;

my $test1 = Cluster->new("One");
print $test1->getClusterName()." ";
my $test2 = Cluster->new("Two");
print $test1->getClusterName()." ";
print $test2->getClusterName();
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#OUTPUT:
one One Two
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Have a happy Perl journey :)...