Add Field to WordPress Register Form

I don’t know about the plugin you are using, but the way to add custom fields to the registration form and saving it to user profile is by hooking into register_form action hook to output the field and then using the appropriate filter/action hook to update/save user profile.

Your case, for example:

<?php
// output the form field
add_action('register_form', 'ad_register_fields');
function ad_register_fields() {
?>
    <p>
        <label for="firstname"><?php _e('First Name') ?><br />
        <input type="firstname" name="firstname" id="firstname" class="input" value="<?php echo esc_attr($_POST['firstname']); ?>" size="25" tabindex="20" />
        </label>
    </p>
<?php
}

// save new first name
add_filter('pre_user_first_name', 'ad_user_firstname');
function ad_user_firstname($firstname) {
    if (isset($_POST['firstname'])) {
        $firstname = $_POST['firstname'];
    }
    return $firstname;
}
?>