Clear Cache for Cart

1. **For WP Super Cache**: The WP Super Cache cache clearing function `wp_cache_clear_cache()` clears the entire page cache. If the cart count is part of a cached page, clearing the cache should ensure it updates when an item is added or removed.

2. **For W3Speedster**: You can use `w3tc_pgcache_flush()` to clear the page cache. However, if the cart count is generated dynamically (like via AJAX or after an item is added), you may need to ensure the cache is purged properly for the section displaying the cart count.

Here’s how you can update your function:

add_action('woocommerce_add_to_cart', 'clear_cache_after_cart_update', 10, 1);
add_action('woocommerce_remove_cart_item', 'clear_cache_after_cart_update', 10, 1);

function clear_cache_after_cart_update($cart_item_key) {
    // Clear WP Super Cache cache
    if (function_exists('wp_cache_clear_cache')) {
        wp_cache_clear_cache();
    }
    
    // Clear W3Speedster cache
    if (function_exists('w3tc_pgcache_flush')) {
        w3tc_pgcache_flush();
    }
    
    // Optional: Clear cart count-related cache if it's cached separately
    if (function_exists('wp_cache_delete')) {
        // Assuming the cart count is stored in a transient or object cache
        wp_cache_delete('woocommerce_cart_count');
    }
}

### Key Notes:
– **Cart Count Transient**: If the cart count is stored as a transient or object cache (e.g., `woocommerce_cart_count`), it's a good idea to delete it explicitly. This ensures the cart count gets refreshed when a change happens.

– **AJAX-based Updates**: If the cart count is updated using AJAX, make sure the cache is cleared only when necessary and that the JavaScript is properly triggering the cart count update after an AJAX request.

This should help ensure the cart count is updated correctly after any addition or removal from the cart.