Consider:

#! perl -w scipt

use strict;
use warnings;

use Data::Dumper 'Dumper';

my @file_names;

my $dir = "csv/";

opendir( DIR, $dir ) || die "can't opendir $dir: $!";
@file_names = grep { /^.*.csv/ && -f "$dir/$_" } readdir(DIR);

closedir DIR;

my $File_number = 0;
my @place_type;
my @place_name;
my @place_tag;
my @reglookup;

open( DIR_A_prep, "$dir/$file_names[0]" )
  || die "can't open file $file_names[0]: $!";

while (<DIR_A_prep>) {

    if ( $_ =~
m/^([^,]+),([^,]+),\d+,\d+,\d+,\d+,,\d+,\d+,\d+,\d+,,\d+,\d+,\d+,\d+/
      )
    {

        my $place_tag_string = $1 . "_" . $2;

        my $reglookup_tag_string = $1 . "," . $2;

        print
"\n>>>>>>>>>>>>>>>>>>>>>>\n$reglookup_tag_string\n>>>>>>>>>>>>>>>>>>>>
+>>.\n";

        $place_tag_string =~ s/ /\_/g;

        #   $place_tag_string  =~ s/\\s/\_/g;
        $place_tag_string =~ s/\"//g;

        $reglookup_tag_string =~ s/ /\\s/g;
        $reglookup_tag_string =~ s/\,/\\,/g;
        $reglookup_tag_string =~ s/\"/\\"/g;
        $reglookup_tag_string =~ s/\(/\\(/g;
        $reglookup_tag_string =~ s/\)/\\(/g;
        $reglookup_tag_string =~ s/\&/\\&/g;

        push( @place_type, $1 );
        push( @place_name, $2 );
        push( @place_tag,  $place_tag_string );
        push( @reglookup,  $reglookup_tag_string );

    }
}
close DIR_A_prep;

my $array_position = -1;

foreach (@place_tag) {

    next unless defined($_);

    $_ =~ s/ /\\s/g;

    $array_position++;


    print "\n@@@@@@@@@@\n\n$reglookup[$array_position]\n\n";

    foreach my $file_name (@file_names) {
        next unless defined($file_name);

        my $rl = $reglookup[$array_position];
        my $pt = $place_tag[$array_position];

        $File_number++;
        print "\n\n\n\n$rl, $dir, $file_name, $File_number, $pt\n\n\n\
+n";
        print_out_files( $rl, $dir, $file_name, $File_number, $pt );
    }
}


sub print_out_files {
    my %files;
    my ( $reglookup, $dir, $file_name, $File_number, $place_tag ) = @_
+;
    my $Is_it_printed = 0;

    $files{Outfile}   = "Result/Result_$place_tag.txt";
    $files{Outfile_B} = "Result/Result_B_$place_tag.txt";

    open( OUTFILE, "+>", $files{Outfile} )
      || die "Can't create output file $files{Outfile}: $!";
    open( OUTFILE_B, "+>", $files{Outfile_B} )
      || die "Can't create output file $files{Outfile_B}: $!";

    print "$file_name\n";
    open( DIR_A, "<$dir/$file_name" ) || die "can't open file $file_na
+me: $!";

    my $Sex_specificity;

    while ( my $line = <DIR_A> ) {
        my %cat;
        my $Cause_of_death_string;
        my $Cause_of_death_ICD;

        if ( $File_number == 1 ) {
            $files{First}{name} = $file_name;

            if ( $line =~ /^Statistic,([^,]*),,*/ ) {
                $files{First}{statistic} = $1;
                print OUTFILE_B "\n$files{First}{statistic}";
            }
            if ( $line =~ /^Age Group,([^,]*),,*/ ) {
                $files{First}{age_group} = $1;
                print OUTFILE_B "\n$files{First}{age_group}";
            }
            if ( $line =~ /^Period,([^,]*),,*/ ) {
                $files{First}{period} = $1;
                print OUTFILE_B "\n$files{First}{period}";
            }
            if ( $line =~ /^Scale,"{0,1}([^,]*)"{0,1},/ ) {
                $files{First}{scale} = $1;

                #    print OUTFILE_B "\n$files{First}{scale";
            }
        } else {
            $files{Current}{name} = $file_name;
            fail( 'statistic method', %files )
              if $line =~ /^Statistic,([^,]*),,*/
              and $files{First}{statistic} ne $1;
            fail( 'age group', %files )
              if $line =~ /^Age Group,([^,]*),,*/
              and $files{First}{age_group} ne $1;
            fail( 'time period', %files )
              if $line =~ /^Period,([^,]*),,*/ and $files{First}{perio
+d} ne $1;
            fail( 'time period', %files )
              if $line =~ /^Scale,"{0,1}([^,]*)"{0,1},/
              and $files{First}{scale} ne $1;
        }

        if ( $_ =~ /^Go to SHA,,MALES,,,,,FEMALES,,,,,PERSONS,,,/ ) {
            $Sex_specificity = "Both";
        } elsif ( $_ =~ /^Go to SHA,,MALES,,,\n/ ) {
            $Sex_specificity = "Males";
        } elsif ( $_ =~ /^Go to SHA,,FEMALES,,,\n/ ) {
            $Sex_specificity = "Females";
        }

        if ( $_ =~ /^Indicator\,\"{0,1}(.+)\"{0,1},,/ ) {
            print OUTFILE $1 . "\t";

            if ( $line =~
                /^Indicator\,\"{0,1}([^(]*)(\([^)]*\))\"{0,1}\:\"{0,1}
+,,/ )
            {
                $Cause_of_death_string = $1;
                $Cause_of_death_ICD    = $2;
            }
        }

        if ( $_ =~ m/^ENG\,ENGLAND,([^\n]+)/ )
        {    # Match any character except new line
            my @pos_A = split /\,/, $1;

            print "This is pos_A:  @pos_A \n\n";
            assignParams( $cat{EngWales}{$_}, splice @pos_A, 0, 4 )
              for qw(First Second Third);
        }

        if ( $_ =~ m/^$reglookup\,([^\n]+)/ ) {
            my @pos = split /\,/, $1;

            assignParams( $cat{PlaceX}{$_}, splice @pos, 0, 4 )
              for qw(First Second Third);
        }
    }

    if ( defined $Sex_specificity ) {
        print OUTFILE_B <<"CAUSE";


Cause of death\tICD-10\tM\tF\tAll\t Obs\tDSR\tL 95% CI\tU 95% CI\tE&W 
+Obs\tE&W DSR\tE&W L 95% CI\tE&W U 95% CI


CAUSE
    }

    if ( $Sex_specificity eq "Both" ) {

        # Do what needs doing here
    } elsif ( $Sex_specificity eq "Males" ) {

        # Do what needs doing here

    } elsif ( $Sex_specificity eq "Females" ) {

        # Do what needs doing here
    }

    close DIR_A;
    close OUTFILE;
    close OUTFILE_B;
}

sub fail {
    my ( $error, %files ) = @_;

    print
"There is a '$error' inconsistency between $files{Current}name and $fi
+les{First}{name}";
    unlink $files{Outfile};
    unlink $files{Outfile_B};
    exit;
}

sub assignParams {
    my ( $hash, @params ) = @_;

    $hash->{observed}                   = $params[0];
    $hash->{DSRs}                       = $params[1];
    $hash->{Lower_95_Confidence_Limits} = $params[2];
    $hash->{Upper_95_Confidence_Limits} = $params[3];
}
[download]

which is about half the length and includes the following changes:

• cleaned up with PerlTidy using default settings (which includes removing "extra" blank lines)
• use a hash for the file name related variables
• use a hash for the "cat" parameter related variables
• use three parameter open
• remove unused global variables
• make all variables as local as possible
• avoid the same name for scalar and array variables
• use $var++ rather than $var = $var + 1
• refactor error reporting code into a fail sub
• use elsif as appropriate to indicate exclusive if clauses
• remove "one use" variables
• refactor parameter extraction code into a assignParams sub
• refactor repeated print code into a single if ( defined $Sex_specificity ) controlled print

The "bug" may be due to the exit (now in sub fail) btw.