Drip-feeding us bits of information isn't really appreciated. Please read "How do I post a question effectively?" and "Short, Self-Contained, Correct Example".

As it stands, we don't know anything about the client side beyond "sends data using an HTTP write"; we don't know what "/home/app_scripts/process.input" does; and "PIDfile" is a complete mystery (you open, write, close, and never use again).

Here's a scenario that works. It does the sort of things that you say you want to achieve. It doesn't generate any error or warning messages. Adapt to your needs.

ken@titan ~/tmp/pm_11151665_io_socket
$ ls -l
total 4
-rwxr-xr-x 1 ken None 387 Apr 15 13:51 client.pl
-rwxr-xr-x 1 ken None  50 Apr 15 11:57 process.sh
-rwxr-xr-x 1 ken None 464 Apr 15 13:58 server.pl
-rw-r--r-- 1 ken None 200 Apr 15 14:01 test.log
[download]

client.pl:

#!/usr/bin/env perl

use strict;
use warnings;

use IO::Socket;

my $client = IO::Socket::INET::->new(
    Proto => 'tcp',
    PeerAddr => 'localhost',
    PeerPort => 55555,
) || die "Can't open client socket: $IO::Socket::errstr";

for my $send_num (1 .. 5) {
    $client->print(<<"EOT");
GET /DATA_String/{send_num = $send_num}HTTP/1.1
host
connection
user-agent

EOT
    sleep 5;
}
[download]

server.pl:

#!/usr/bin/env perl

use strict;
use warnings;

use IO::Socket;

my $server = IO::Socket::INET::->new(
    Proto => 'tcp',
    LocalPort => 55555,
    Listen => SOMAXCONN,
    ReuseAddr => 1,
) || die "Can't open server socket: $IO::Socket::errstr";

my $client = $server->accept();
$client->autoflush;

while (defined(my $read = <$client>)) {
    chomp $read;

    if (index($read, 'GET /DATA_String/') == 0) {
        system('sh', 'process.sh', $read);
    }
}
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process.sh:

#!/usr/bin/env sh

echo -e "$*" >> test.log 2>&1
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Start server.pl then start client.pl. These will finish after ~25secs. Now check the log.

$ cat test.log
GET /DATA_String/{send_num = 1}HTTP/1.1
GET /DATA_String/{send_num = 2}HTTP/1.1
GET /DATA_String/{send_num = 3}HTTP/1.1
GET /DATA_String/{send_num = 4}HTTP/1.1
GET /DATA_String/{send_num = 5}HTTP/1.1
[download]

See also: IO::Socket; IO::Socket::INET; and, perlipc. The perlipc page has a lot of content; if you don't want to read it all, concentrate on "TCP Clients with IO::Socket" and "TCP Servers with IO::Socket" (which should answer questions you might have about my code).