Plaid's Auth product is billed under the one-time fee pricing model. This means that you will incur charges for the Auth product whenever an Item is initialized with Auth. This occurs when an Item has been successfully created by Link and the Auth product was included in Link's product array. An Item is considered successfully created if an access_token was generated.
If the Item was not initialized with the Auth product at the time of creation, it can also be initialized later by calling the product endpoint belonging to the Auth product - /auth/get - on the Item.
For one-time fee products like Auth, it does not matter how many API calls are made for the Item; the cost is the same regardless of whether the product's endpoints are called many times or zero times.
Auth requires the item to be healthy. Auth will not be billed if it is called on an item that is in an unhealthy state. Note that you can check the health status of an item using the /institutions/get_by_id endpoint in the Plaid API.