The public API is going to do all the adding/removing/sorting/ and reordering for you. It will (or should) have public endpoints that represent each one of those pieces of functionality. Say you were using the Todoist API:
To add an item to your list, you would add a request in Postman for their Create A New Task endpoint. You would then use Postman to build the request body, add appropriate headers, and authorize.
When you execute the request, it will run the endpoint in Todoist and create a new item for your project in your account. Todoist handles the database side and backend logic for you, you are simply generating and executing the request from Postman.
You can then add other requests into your collection like Update a Task (for priority/sorting) or Delete a Task.
A feature that makes Postman real neat is the ability to execute a collection via the Collection Runner, firing off requests one after another. So with the click of a button, you can create a new task, set the priority, then delete it.
By utilizing tests on your requests, you can verify the data was successfully created, updated, and deleted.
Hope this helps! Good luck!