How to add CSS to this cart after hover?

not sure what you are wanting to do but the CSS for the cart is:

.ast-woocommerce-cart-menu .ast-cart-menu-wrap .count, .ast-woocommerce-cart-menu .ast-cart-menu-wrap .count:after {
  border-color: #0170B9;
  color: #0170B9;
}
.ast-woocommerce-cart-menu .ast-cart-menu-wrap:hover .count {
  color: #ffffff;
  background-color: #0170B9;
}

Found it!

Thanks @tugbucket! It was close to what you replied. But that led me to the solution that I needed. This is the code that I was looking for:

.widget_shopping_cart_content{
   background: #fff;               
 }

Thanks again!