use Modern::Perl;

my $dirname = 'c:/data/eps';
opendir( DIR, $dirname );
my @files = readdir(DIR);
foreach my $file (@files) {
    my $file_path = "$dirname/$file";
    say "File path: $file_path";
}
closedir DIR;
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File path: c:/data/eps/.
File path: c:/data/eps/..
File path: c:/data/eps/agrafort.jpg
File path: c:/data/eps/alazif
File path: c:/data/eps/ARF.jpg
File path: c:/data/eps/ARF.png
...
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#!/usr/bin/perl
use 5.014;                  # use 5.014 is effectivly the same as use 
+strict.  Do it. 
use File::Spec;             # you can't use File::Spec and components 
+until you've
                            # imported the module and its components
my ($FILE, @FILES);         # strict requires my; more important, it w
+ill tell you about many goofs
my $dirname="...
[download]

Update: fixed missing words in comment.

@txt_files = glob qq("C:\\Users\\Me\\Desktop\\Cluster1\\*.txt");
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Unlike readdir, glob returns the full path in the same format as you passed as an argument: absolute path in yields absolute path out; relative path in (relative to the current working directory of perl which you can change with chdir) yields relative path out.

Note that if (and only if) your $dirname contains spaces, then the double quotes are required. You don't have to put them around the whole path-with-wildcards, replacing every single space with qq(" ") will work, too.

This works for me:

use File::Util;
my $f = File::Util->new(); 
my @files = $f->list_dir( "C:\\book", qw( --no-fsdots --with-paths ) )
+;
[download]

From the command line: (demonstrating that "/" works fine on windows)

$ perl -MFile::Util -e 'print join "\n", File::Util->new->list_dir("C:
+/book", qw( --no-fsdots --with-paths ))' 

C:/book/Generic_User_Guide.pdf
C:/book/Quick_Guide.pdf
[download]

#!/usr/bin/perl -l

use strict;
use warnings;

my $dirname = shift @ARGV;
opendir( DIR, $dirname );

my(@files) = readdir(DIR);
print "File path: ";
foreach my $file (@files) {
    my $file_path = "$dirname/$file";
    $file_path =~ s/\\|\//\\\\/g;
    print $file_path;
}
closedir DIR;
[download]

Am adding just a "kobo" suggestion, using your OP and the modification provided by ww to your script.

Anonymous Monk says:
"..D:\Prog\Perls is the location where I execute my code.."

In that case, you can get your desired result using chdir. {Updated}
Like so:

#!/usr/bin/perl
use warnings;
use strict;
use File::Spec;

my $dirname = "C:\\Users\\Me\\Desktop\\Cluster1";
my @file_to_use_later;    ## for use later
chdir $dirname;           ## add this to OP script

opendir DIR, $dirname or die "can't open directory: $!";
my @FILES = readdir(DIR);
foreach my $FILE (@FILES) {
    my $fil_path = File::Spec->rel2abs($FILE);
    print "File path:", $fil_path, $/;
    push @file_to_use_later, $fil_path if -f $fil_path;    ## check fi
+les contents later
}
closedir DIR or die "can't close directory: $!";

## test the files stored in the array variable later

for my $filename (@file_to_use_later) {
    print $filename, $/;
    open my $fh, '<', $filename or die "can't open: $!";
    while (<$fh>) {
        chomp;
        print $_, $/;
    }
    close $fh or die "can't close file:$!";
}
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    $abs_path = File::Spec->rel2abs( $path ) ;
    $abs_path = File::Spec->rel2abs( $path, $base ) ;
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D:\Prog\Perls\doc.txt
D:\Prog\Perls\volt.txt
D:\Prog\Perls\holiday.txt
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C:\Users\Me\Desktop\Cluster1\doc.txt
C:\Users\Me\Desktop\Cluster1\volt.txt
C:\Users\Me\Desktop\Cluster1\holiday.txt
[download]