Below I made two small changes that may do what you want. If not, you've been given all the concepts; try yourself!

use strict;
use warnings;

my $file = "z.txt";
open my $fh, "<", $file or die "Unable to open $file: $!";
my ($p_val, $m_val);
{ local $/ = '\n\n\n';
  while (<$fh>) {
     foreach (split/\n/) {
        if (s/\bProcessing\s\d+\.tx\.\d+: //) {
            print "$_\n";
        } elsif (s/\bPhrase: //) {
            s/"//g;
            $p_val = $_;
        } elsif (/^\s\s(.+)/) {
            $m_val = $_;
            if (defined $p_val) {
               print "\t$p_val\t$m_val\n";
               undef $p_val;   ### moved undef here
            } else {           ### new else condition
               print "\t\t$m_val\n";
            }
        }
     }
  }
}
close $fh;
__END__
Pulmonary embolism at the time of hip replacement
        Pulmonary embolism        1000 D0076131:PULMONARY EMBOLISM Dis
+ease or Syndrome
        of hip replacement        1000 D0554893:HIP REPLACEMENT (STATU
+S POST HIP REPLACEMENT) Finding
        of hemorrhage     1000 D0046004:HAEMORRHAGE (BLEEDING) Finding
                  1000 D0046011:HAEMORRHAGE NOT OTHERWISE SPECIFIED (H
+EMORRHAGE NOT OTHERWISE SPECIFIED) Finding
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