Hmmm.....I don't know that this has anything to do with the Tape Cue. I once had tape explained to me thusly: You have 4 tracks of tape in parallel, adjacent to one another. If you record something really hot (loud), a few things happen - the actual audio information is pushed towards the edges of those tracks, and you get natural compression from the tape. So, I believe (based on my experience) if you record something too hot, and then record over it, some of the original info remains on the tape (at the outside edges), giving you little ghosty remnants of the click (in your case). In fact, I think that audio can actually spil over onto neighboring tracks.
The solution, therefore, would be to record your click, and anything else you want to later tape over, at a lower volume. [just boost it accordingly at playback when you need it]
(someone please correct me if this is all utter bunk)
Are you recording with the direct method or the Buss L/Buss R method? It shouldn't make a difference regarding your problem, I don't think, but I'm just curious. If you are unfamiliar with the Buss method, try it out. Once you understand what's going on, it can make things easier - you don't have to plug and unplug cables constantly, adjust trim and faders as much, etc. But, I don't think that should have anything to do with your problem.