Step 1: You will need to set up an override customer password in the security settings:
Settings > Security > General
Scroll to and enter the ‘Customer Override Password’
Step 2: Close out of the admin console completely, and open a new browser window. Here log in as the customer for the cart you need to delete.
Step 3: View the customers cart (if no link exist use the direct URL /shopcart.aspx). When cart page is visible you can either click the “clear cart items” to remove all items from the cart, or you can click the “remove button” to clear only single items from the cart.
Step 4 : Logout of store front and close the browser.