You're forgetting about the newlines that you get with the <STDIN> operators. Your example would be better written as:

use strict;
use File::Spec; 
print "File Name?\n";
my $ToBeDeleted = <STDIN>;
chomp $ToBeDeleted;
open(IN, $ToBeDeleted) or die " The file $TheFileToBeDeleted .....Miss
+ing In Action.\n";
my @arr = <IN>;
chomp @arr;
print "Directory where files are located?\n";
my $Path = <STDIN>;
chomp $Path;
foreach my $file (@arr)
{
 my $x=File::Spec->catfile($Path, $file); 
 unlink $x;
}
[download]

You have a few problems other than the newlines: for one thing, you put quotes around your variables, interpolating them into strings (why do people do this?). This will make your @arr into a space-separated string, which is not what you want. And you're not processing each of your files separately through catfile.