Just after Neo is in the Constuct for the first time,(when he yells, get me out, I want out), he is lying on a chair with a plug in the back of his head, he tries to sit up only to be pushed down by Trinity who then takes the plug out of his head. if you look carefully when he sits up there is no plug.
Yes, the actors all wanted a chance to have a spike shoved into the back of their heads! To have it glued on, and then removed so quickly would have caused even bigger slipups...