Get ‘Headers already sent’ error for the plugin I am creating when I try to login

Your syntax is wrong. You have a closing PHP tag where there is no opening PHP tag at the end of the script. This means you are creating output before the headers are being sent.

Change this:

function prodmanager_html_page() {
?>

<div>

<h2>Product Manager</h2>
<!--usual HTML and PHP Code--> 
</div>
?>

To this:

function prodmanager_html_page() {
    ?>
    <div>
        <h2>Product Manager</h2>
        <!--usual HTML and PHP Code--> 
        </div>
    <?php
}

You do not need a closing PHP tag at the end of your script. This is a good practice to help safeguard against stray whitespace after the closing tag which will give WP a white screen of death.