My public website has a 'cart' link in the header
We have coded WooCommerce Private Store to hide the cart icon from the header of the most popular WordPress themes. If it’s still appearing in your theme, send us a support request and we’ll help you hide it.
Alternatively, if you are comfortable editing CSS, you could add some code to your theme or under the Appearance → Customise → Additional CSS box. When the stock is locked, there is a woocommerce-locked class added to the <body> tag. You can use this to target elements you want to hide when the store is locked. E.g.
.woocommerce-locked .main-menu .cart-icon {
display: none !important;
}
If searching the knowledge base hasn't answered your question, please contact support.