This uses HTML::TokeParser

#!/bin/perl5

use strict;
use warnings;
use HTML::TokeParser;

my $file = 'map2004.html';
my $tp = HTML::TokeParser->new($file)
  or die "Couldn't read html file: $!";

# start tag, attrib, value  
my ($s_tag, $s_attrb, $s_value) = qw(div class menu);
# end tag
my ($e_tag) = 'h6';
my $max = 20;

my $count;

my $start;  # flag
# typo fixed
my %data;  # hash to hold output

while ( my $tag = $tp->get_token ) {
  
  next if
    $tag->[0] eq 'S' and
    $tag->[1] eq $s_tag and 
    exists $tag->[2]->{$s_attrb} and 
    $tag->[2]->{$s_attrb} eq $s_value and
    ++$start;
    
  next unless $start;
  
  last if
    $tag->[0] eq 'S' and
    $tag->[1] eq $e_tag;
    
  if (
    $tag->[0] eq 'S' and
    $tag->[1] eq 'a' and
    exists $tag->[2]->{href}
  ){
      my $href      = $tag->[2]->{href};
      my $link_text = $tp->get_trimmed_text('/a');
      $data{$href}  = $link_text;
      $count++;
      last if $count == $max;
  }
}
for my $key (sort keys %data){
  print "$key -> $data{$key}\n";
}

# ["S",  $tag, $attr, $attrseq, $text]
# ["E",  $tag, $text]
# ["T",  $text, $is_data]
# ["C",  $text]
# ["D",  $text]
# ["PI", $token0, $text]
[download]

update

"...specify in some simple way what data to extract..."

That's trickier because it depends on 'what data'.
I've always found it relatively easy to adapt the above type of script.

update 2

Fixed typo.