After considering hdb and Kenosis's advice, when I need to iterate over an array I am dropping terms from, I usually use an explicit index and iterate backwards over the indices. Note that you already need to use $delele (which I assume should be $delete) to track the index. As long as you are already tracking it, you might as well include that in your control structure. So maybe your code would read (untested):

 
use strict;
use warnings;
use File::Copy;

my @files = glob("*.hml");
my @DIR = glob ("*/*.hml");


my @array = ('', grep -d, glob '*');
print "Here's DIR @DIR";
my $argsucess = 0;
foreach my $el (@ARGV){
    print "\n\n\nheres el $el";
    foreach my $delele (reverse 0 .. $#DIR){
       my $element = $DIR[$delele];
       print "\n\n\nHERE'S ELE $element";
       print "\n\n\n\n\n\n HERE'S DIR SO FAR @DIR\n\n\n";
        my $re1='((?:[a-z][a-z]+))'; # Non-greedy match on filler
        

        my $re=$re1;
        my $path;
        if ($element =~m/$re/is){
            $path = $1;
        }
       
        if ($el eq $path){
            
            splice(@DIR, $delele, 1);
            $argsucess++;
             next;     
        }
    }
    
}
[download]

The benefit to reverse traversal is that you don't interfere with the indices for future iterations by splicing this one out.

You might also consider using qr// for your regular expression (see Regexp Quote Like Operators). I'm a little unclear on why you aren't just using a fixed regular expression in if ($element =~m/$re/is){, nor why you are using the my $re=$re1; misdirection. Also, by initializing my $path; to undef instead of empty string (my $path = '';), you will likely get a number of unnecessary warnings from if ($el eq $path){.