Actually, I have started to answer this question twice and deleted my answer both times. The reason lies in trying to answer the procedure portion of your question, I was not satisfied with my answer. As a general rule there is no reason to remove any of your birds food in order to give it a treat. There are no set procedures to giving treats to your bird.
I think that you may be having some sort of problem in relation to treats and probably in conjunction with training your bird. I would guess that the bird is refusing the treats that you are offering while you are attempting to work on training your bird. But I don't know the exact nature of your issue which makes it difficult to give you any beneficial suggestions on how to proceed.
If your bird is not accepting the treats that you are offering the problem could be what you are offering as a treat, such as the treat not being enough of a high value of a food or even that it does not recognize the treat as food. It could be a matter of what the birds daily diet consists of, which may be devalueing the treat. The problem could be in your timing of the training sessions, resulting in the bird not being hungry at that time.
I have always kept food in my birds food dish except for while the bird is sleeping at night and have never had any issue with my birds accepting treats when I offer them. If you will describe to us the problem that you are experiencing, we can offer more accurate suggestions to you to eliminate the problem.