Second, I find it hard to believe you can delete files on this host but that you can't create any. On most systems, your home directory or the /tmp directory is writeable by you. If you truly can't create any files anywhere then your script is never going to work because your own code is creating a file.

Third, CGI::Carp can't display compile time errors to the browser without a little help. For that you would need to bypass the web server log. See this for details. It assumes that you really can create a file.

Fourth, you should still be able to validate that the script compiles even if you don't have perl locally or a shell account on the host. Create a new script that executes a system call to perl -wc of the first script. Since you are unlinking files and opening pipes to external mail programs - I don't see why this wouldn't work.

I do hope that you can get things to work. If none of these other pieces of information are helpful then try a completely different approach. Start out with a working script that doesn't nothing more than prints "hello, world" to the browser screen. Next add on a single bit of information and test it. Continue this process until you find exactly what the problem is or you get a working solution.

Update:
I have taken the liberty of re-writing your code to something I know compiles and corrects some of the logic problems of your own. I say some because you have a race condition that you will have to address on your own if it is a real issue. Use this at your own risk as it comes with no guarantees.

#!/usr/bin/perl
use strict;
use warnings;
use CGI qw/:standard/;
use CGI::Carp qw(fatalsToBrowser);

my $prog = '/usr/lib/sendmail'; 
my $to   = 'Foo@nowhere.com';
my $from = 'Bar@somewhere.com';
my $file = 'report.txt';

my $num = 0;
if ($ENV{QUERY_STRING} eq 'new' && -e $file) {
    unlink $file or die "Unable to remove '$file': $!";
}
else {
    open(my $fh, '<', $file) or die "Unable to open '$file' for readin
+g: $!";
    $num = <$fh>;
    chomp $num;
}
my ($col, $n) = (2, 0);
for my $i (1 .. $col) {
    $n = $i + $num;
    open (my $mail, '|-', $prog) or die "Unable to open a pipe to '$pr
+og': $!";
    gen_message($mail, $to, $from);
}

open(my $fh, '<', $file) or die "Unable to open '$file' for writing: $
+!";
print $fh $num + $col;
close $fh;

print header, start_html($num), $num, end_html();

sub gen_message {
    my ($mail, $to, $from) = @_;
    print $mail "Content-Type: text/plain; charset = windows - 1251\n"
+;
    print $mail "Subject: Text, Text!\n";
    print $mail "To: $to\n";
    print $mail "From: $from\n\n";
    print $mail "Hello, the message text itself..\n";
    print $mail "\n\n";
}
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