Ah ok that makes sense now. No you won’t be able to have a private task within a public project. You could only add some people with comment only access but they would still be able to see the task.
How to overcome this: You could create a task just with the task name and then in the description field add the Asana link of the task you create in the private board that the others do not have access to.
If you are interested in this functionality I recommend upvoting here: Private and Hidden Tasks/Sections in Public Projects