Emeğin, yardımların için teşekkür ederim.
Kodu kullanmaya çalıştığımda bir kaç hata aldım.
Araştırmalarım sonucu amacımı birebir karşılayan bir fonksiyon buldum.
İhtiyacı olan arkadaşlar tema lokasyonunuzun içinde bulunan "functions.php" dosyasının en altına ekleyerek kullanabilirsiniz.

Kaynak

// 1. Display field on "Add new product category" admin page
 
add_action( 'product_cat_add_form_fields', 'bbloomer_wp_editor_add', 10, 2 );
 
function bbloomer_wp_editor_add() {
    ?>
    <div class="form-field">
        <label for="seconddesc"><?php echo __( 'Second Description', 'woocommerce' ); ?></label>
      
      <?php
      $settings = array(
         'textarea_name' => 'seconddesc',
         'quicktags' => array( 'buttons' => 'em,strong,link' ),
         'tinymce' => array(
            'theme_advanced_buttons1' => 'bold,italic,strikethrough,separator,bullist,numlist,separator,blockquote,separator,justifyleft,justifycenter,justifyright,separator,link,unlink,separator,undo,redo,separator',
            'theme_advanced_buttons2' => '',
         ),
         'editor_css' => '<style>#wp-excerpt-editor-container .wp-editor-area{height:175px; width:100%;}</style>',
      );
 
      wp_editor( '', 'seconddesc', $settings );
      ?>
      
        <p class="description"><?php echo __( 'This is the description that goes BELOW products on the category page', 'woocommerce' ); ?></p>
    </div>
    <?php
}
 
// ---------------
// 2. Display field on "Edit product category" admin page
 
add_action( 'product_cat_edit_form_fields', 'bbloomer_wp_editor_edit', 10, 2 );
 
function bbloomer_wp_editor_edit( $term ) {
    $second_desc = htmlspecialchars_decode( get_woocommerce_term_meta( $term->term_id, 'seconddesc', true ) );
    ?>
    <tr class="form-field">
        <th scope="row" valign="top"><label for="second-desc"><?php echo __( 'Second Description', 'woocommerce' ); ?></label></th>
        <td>
            <?php
          
         $settings = array(
            'textarea_name' => 'seconddesc',
            'quicktags' => array( 'buttons' => 'em,strong,link' ),
            'tinymce' => array(
               'theme_advanced_buttons1' => 'bold,italic,strikethrough,separator,bullist,numlist,separator,blockquote,separator,justifyleft,justifycenter,justifyright,separator,link,unlink,separator,undo,redo,separator',
               'theme_advanced_buttons2' => '',
            ),
            'editor_css' => '<style>#wp-excerpt-editor-container .wp-editor-area{height:175px; width:100%;}</style>',
         );
 
         wp_editor( $second_desc, 'seconddesc', $settings );
         ?>
      
            <p class="description"><?php echo __( 'This is the description that goes BELOW products on the category page', 'woocommerce' ); ?></p>
        </td>
    </tr>
    <?php
}
 
// ---------------
// 3. Save field @ admin page
 
add_action( 'edit_term', 'bbloomer_save_wp_editor', 10, 3 );
add_action( 'created_term', 'bbloomer_save_wp_editor', 10, 3 );
 
function bbloomer_save_wp_editor( $term_id, $tt_id = '', $taxonomy = '' ) {
   if ( isset( $_POST['seconddesc'] ) && 'product_cat' === $taxonomy ) {
      update_woocommerce_term_meta( $term_id, 'seconddesc', esc_attr( $_POST['seconddesc'] ) );
   }
}
 
// ---------------
// 4. Display field under products @ Product Category pages
 
add_action( 'woocommerce_after_shop_loop', 'bbloomer_display_wp_editor_content', 5 );
 
function bbloomer_display_wp_editor_content() {
   if ( is_product_taxonomy() ) {
      $term = get_queried_object();
      if ( $term && ! empty( get_woocommerce_term_meta( $term->term_id, 'seconddesc', true ) ) ) {
         echo '<p class="term-description">' . wc_format_content( htmlspecialchars_decode( get_woocommerce_term_meta( $term->term_id, 'seconddesc', true ) ) ) . '</p>';
      }
   }
}