You should be able to either remove the box or to put a barrier between the box and the cage, so the birds can not get into it. Budgies will continue to breed as long as conditions are good, but after the first couple of clutches, you will find that the babies are not as healthy because the parents are getting worn out. The parent's health will suffer and the babies will not survive.
You put the picture on the web and then insert the picture here. In order to get the button to insert a picture, you have to either start a new post or hit the reply button instead of using the quick reply to a post.