Here is a version which may help. You do not need the next LINE bit, and the C-style loop hides a bug - you never test for the last item in @tmpArray (BB_B06).

my @arrayOfVals;
open(my $tmp, "<", "/Users/bioinformatics/Desktop/NC_001903.gff.txt") 
+or die "Could not open $!";
while (<$tmp>) {
    chomp;
    next if /^#/; # discard header, unneeded and will interfere
    push(@arrayOfVals, $_);
}
close($tmp);

my %hashOfKeys;
open (my $arrVal, "<", "/Users/bioinformatics/Desktop/cp26_dff.txt") o
+r die "Could not open $!";
while (<$arrVal>) {
    chomp;
    $hashOfKeys{$_} = 1;
}
close($arrVal);

my %hash;

for my $k (0 .. $#arrayOfVals -1) {
    if ($arrayOfVals[$k] =~ /.*;locus_tag=(.*)/) {
        next unless $hashOfKeys{$1};
        my $key = $1;
        if ($arrayOfVals[$k+1] =~ /.*;Name=(.*);protein_/) {
            $hash{$key} = $1;
        }
    }
}
print Dumper(\%hash);
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