Some problems with the code shown:

• $self is a class variable, it should be an object (instance) variable.
• Subroutines have unneccessary prototypes.
• use strict and use warnings are missing.
• Module names should be capitalised.

But the main problem is the structural relationship of the child classes to the parent class. Here is one way to do what is wanted:

#! perl
use strict;
use warnings;

#---------------------------------------------------------------------
+---------
{
    package SuperModule;

    sub new
    {
        my ($class, %params) = @_;
        my  $self = { dbh => $params{DB} };
        return bless $self, $class;
    }

    sub print_super
    {
        my ($self) = @_;
        print 'Super module: ', $self->{dbh}, "\n";
    }
}

#---------------------------------------------------------------------
+---------
{
    package MyModule;

    our @ISA = qw( SuperModule );

    sub new
    {
        my ($class, %params) = @_;
        my  $self = SuperModule::new($class, %params);
        $self->{file_name} = $params{'file name'};
        return bless $self, $class;
    }

    sub print_file
    {
        my ($self) = @_;
        print 'My file name from file obj ', $self->{file_name}, "\n";
    }
}

#---------------------------------------------------------------------
+---------
{
    package YourModule;

    our @ISA = qw( SuperModule );

    sub new
    {
        my ($class, %params) = @_;
        my  $self = SuperModule::new($class, %params);
        $self->{block_name} = $params{'block name'};
        return bless $self, $class;
    }

    sub print_block
    {
        my ($self) = @_;
        print 'My block name from block obj ', $self->{block_name}, "\
+n";
    }
}

#---------------------------------------------------------------------
+---------

my %f_params  = (
                    DB           => 'File DB value',
                    'file name'  => 'Lisa',
                );
my %b_params  = (
                    DB           => 'Block DB value',
                    'block name' => 'Pisa',
                );
my $file_obj  = MyModule  ->new(%f_params);
my $block_obj = YourModule->new(%b_params);

$file_obj ->print_super();
$file_obj ->print_file();
$block_obj->print_super();
$block_obj->print_block();
[download]

Output:

13:56 >perl 567_SoPW.pl
Super module: File DB value
My file name from file obj Lisa
Super module: Block DB value
My block name from block obj Pisa

13:56 >
[download]

Hope that helps,

Update: ++tobyink, below, for the improvements to the child class constructors.