How big are the 2 files ? Try

#!/usr/bin/env perl 
use strict;
use warnings;

my %count = ();

my $dir = '/home/nic/Desktop/5-27-17/';
my $file1  =  $dir.'Genomes_used_hant.txt';
open IN,'<',$file1 or die "Could not open $file1 : $!";
while (<IN>){
  chomp;
  $count{$_} = 0;
}
close IN;

my $file2 = $dir.'NRT2.txt';
my $n = 0;
open IN,'<',$file2 or die "Could not open $file2 : $!";
while (my $line = <IN>){
  if ($line =~ /^>(.*)_/){
    ++$count{$1} if exists $count{$1};
  }
  ++$n;
}
close IN;
print "$n lines read from $file2\n\n";

for (sort keys %count){
  if ($count{$_} > 0 ) {
    printf "%-15s  count %d\n",$_,$count{$_};
  }
}
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