stumped with recaptcha verification step

I hate asking for help.. but I've spent I dunno how many hours trying to figure out how to implement the verification step for this form with recaptcha.

I understand the I need define the private key somewhere, but most of the other guides are using "isset" functions and I just don't at which part of the php to do it.

At the moment, the button shows up and is checkable.. but when I click submit, it sends regardless of whether the box was checked. (Just explaining in case that's helpful.)

Thanks, sorry.

<?php

$to = 'example@caraudioinc.com';
$subject = 'CarAudioInc.com Form Message';

$errName = $errEmail = $errMessage = "";
$name = $email = $pNum = $message = "";


if ($_SERVER["REQUEST_METHOD"] == "POST") {
  $name = test_input($_POST["name"]);
  $email = test_input($_POST["email"]);
  $pNum = test_input($_POST["pNum"]);
  $message = test_input($_POST["message"]);

}

function test_input($data) {
  $data = trim($data);
  $data = stripslashes($data);
  $data = htmlspecialchars($data);
  return $data;
}

if ($_SERVER["REQUEST_METHOD"] == "POST") {

    if (empty($_POST["name"])) {
    $errName = "*Name field is required.";
    } else {
    $name = test_input($_POST["name"]);

        if (!preg_match("/^[a-zA-Z ]*$/",$name)) {
      $errName = "Invaild name."; 
        }
    }

    if (empty($_POST["email"])) {
    $errEmail = "*Email field is required.";
        } else {
    $email = test_input($_POST["email"]);

        if (!filter_var($email, FILTER_VALIDATE_EMAIL)) {
      $emailErr = "Invalid E-mail."; 
        }
    }

    if (empty($_POST["message"])) {
    $errMessage = "*Message field is required.";
        } else {
    $message = test_input($_POST["message"]);
    }

$body = <<<EMAIL

Message from: $name

$message


Phone: $pNum
Email: $email

EMAIL;

$header = "From: $email";   

if (!$errName && !$errEmail && !$errMessage ) {
    if($_POST) {



        mail($to, $subject, $body, $header);

        header('Location: /sent');
    }

    }
}
?>


<!doctype html>
<html lang="en">
<head>
<meta charset="utf-8">
<meta http-equiv="X-UA-Compatible" content="IE=edge">
<meta name="viewport" content="width=1152">
<script src="https://www.google.com/recaptcha/api.js"></script>


<!-- Global site tag (gtag.js) - Google Analytics -->
<script async src="https://www.googletagmanager.com/gtag/js?id=UA-100617570-1" ></script>
<script>
  window.dataLayer = window.dataLayer || [];
  function gtag(){dataLayer.push(arguments);}
  gtag('js', new Date());

  gtag('config', 'UA-100617570-1');
</script>

<!--Google meta tags -->
<title>Car Audio Inc | Contact</title>

<!--fb meta tags -->
<link rel="canonical" href="https://www.caraudioinc.com/contact" />
<link href="css/multiColumnTemplate.css" hreflang="en" rel="stylesheet" type="text/css">
<link rel="icon" href="images/favicon.png" type="image/png">
</head>

<body>
<div class="container">

  <header>
     <div class="primary_header">


    <nav class="secondary_header" id="menu">
      <ul>
     <li><a href="/">HOME</a></li>
        <li><a href="about">ABOUT</a></li>
          <li><a href="services">SERVICES</a></li>
        <li class="current"><a href="contact">CONTACT</a></li>
      </ul>
    </nav>

  </header>

 <div class="contactForm">
  <form method="post" action="<?php echo htmlspecialchars($_SERVER["PHP_SELF"]);?>"><br>

    <label for="name">Name&nbsp;&nbsp;</label><span class="error"><?php echo $errName; ?></span><br>
    <input type="text" id="name" name="name" value="<?php echo $name;?>"><br><br>



      <label for="email">Email&nbsp;&nbsp;</label><span class="error"><?php echo $errEmail; ?></span><br>
    <input type="email" id="email" name="email" value="<?php echo $email;?>"><br><br>



      <label for="pNum">Phone Number (optional)<br></label>
    <input type="tel" id="pNum" name="pNum" value="<?php echo $pNum;?>"><br><br>


      <label for="message">Message&nbsp;&nbsp;</label><span class="error"><?php echo $errMessage; ?></span><br>
    <textarea id="message" name="message" style="height:100px"><?php echo $message;?></textarea>

     <div class="g-recaptcha" data-sitekey="6LcR-DQUAAAAAB2cXajVPi7lpFEZ_z8vZrS97ZXo"></div><br>



    <input type="submit" value="Submit">
    </form>


  <footer class="secondary_header footer">
    <div class="copyright">&copy; Car Audio Inc 2020, All Rights Reserved.</div>

  </footer>

</div>
</body>
</html>