It sounds like you do some kind of matrix. The sound comes from the card hitting the coin and/or the coin hitting the table. Use a close-up mat or tablecloth to work on. Take the card and bend it slightly lenght- and sideways, it should now be 'hollow'. This will reduce the sound a bit because a smaller part of the coin will hit the card. That is all you can do about your material, but there is a way to get around that. The easiest and most effective way is vocal misdirection,'patter'. That is you talking to your spectators. If they listen to your voice they won't hear noises, they will listen to you. Patter is important. But that is not the problem i think.
You, my friend, suffer from 'magicians guilt', i am afraid. That happens to us all but it is easy to beat.
We magicians are our own worst enemies when it comes to practise. Just think about it. When you practise you are alone, extremely concentrated, in a quiet place, often looking in a mirror, repeating the same thing over and over and you are not talking to anybody. The exact opposite of what we do when we are performing.
Now about the clicking noise. You know the trick and you know when the sound is going to happen, so subconcious you are waiting for it. You get extra sensitive for that particular click and your brain treats it as a danger. So you get tense. You jam up on the trick because the click is distracting and annoying you.
Well, it isn't you, your coins or your cards that are the problem, it is the lack of background-noise.
When you perform there is background-noise all around you that will drown any clicking and you will be talking to your audience, so nobody will notice a click, and if they do they won't link it to you or your magic, it is just another sound.
Think as the spectator now. This guy(you) is doing some amazing magic with cards and coins and he tells a good story with it. You(spec) think you heard a faint clicking sound. Would you, a spectator who most likely never saw a real magician before, and who knows nothing about gaffs and gimmicks, suspect PK? No, because you don't know what that is, and even if you did, paper and copper are not magnetic so you have no reason to suspect the props. Subconciously you will assume it is somebody stirring his coffee or taking out his keys, because everybody knows cards don't make a clicking noise and you will forget it immediately. Sounds crazy but it works like that.
Now back to practise.
Find a tape of a camp-fire and play it loud enough to camouflage the click-sound. Use speakers, it doesn't work with earbuds. Start practising as relaxed as you can, easy and slow. Concentrate on the handling. After a while the click will go up in the sound of the crackling fire and you won't hear it anymore. Then you will improve very quick since your brain will no longer be alert for that specific click, it is no longer a treat. You will get more confidence and handle your props better. Then start again and this time say out loud what you are doing, this will get you used to speaking and sleight-of-hand at the same time. Once you get the trick right think of some patter to go with it. I hope you can do something with this.