As people have said is because your keys don't exists.
I've neatened up your code a bit, to make it more readable.
Some pointers:
- use English: This lets you use $INPUT_RECORD_SEPARATOR instead of $\ or $/ I can't even remember which one, I only knew from your input file
- use Data::Dumper: This lets you dump variables to see what's inside them
- use lowercase keys (lc) this ensures case isn't an issue


The output I get is:
john 10 50
mathew 27 40


Code:

#!/usr/bin/perl
use warnings;
use English q/-no_match_vars/;
use strict;
use Carp qw/croak carp/;
use Data::Dumper;

my $file_one_ref;
my $file_one_fn = shift or croak "No input file specifed for file one"
+;
my $file_two_fn = shift or croak "No input file specifed for file two"
+;

open my $file_one_fh, '<', $file_one_fn
  or croak "Could not open filename [$file_one_fn] : ", $OS_ERROR;

while (<$file_one_fh>) {
    $file_one_ref->{ lc $1 } = $2 if / \A (\w+) \s+ (\w+) /xms;
}

# You can do the above input like this but it's not as efficient:
# map { $file_one_ref->{ lc $1 } = $2 if / (\w+) \s+ (\w+)/xms; } <$in
+put_fh>;

# Have a look at the keys read in:
print Dumper($file_one_ref), "\n\n";

open my $file_two_fh, '<', $file_two_fn
  or croak "Could not open filename [$file_two_fn] : ", $OS_ERROR;

$INPUT_RECORD_SEPARATOR = '>';

while ( my $line = <$file_two_fh> ) {

    # I reserve s{}{}xms; for multi line expressions
    next if ( $line !~ s/ \A (\S+) \s+ (?= \d ) //xms and !exists $fil
+e_one_ref->{ lc $1 } );

    # why not just use a match? is $name going to contain numbers

    my $name = lc $1;
    my $numbers = [ grep { $_ } split /\D+/, $line ];
    my $one_number = $numbers->[ $file_one_ref->{$name} - 1 ] || next;
    if ( $one_number >= 20 ) {
        print join " ", $name, $file_one_ref->{$name}, $one_number, "\
+n";
    }
}

close $file_one_fh
  or croak "Could not open filename [$file_one_fn] : ", $OS_ERROR;
close $file_two_fh
  or croak "Could not open filename [$file_two_fn] : ", $OS_ERROR;

__END__
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