Troubleshooters should understand that Friend Computer would never make a clone a mutant
I'm putting this concern first because if you don't have it, I think that mutants in Paranoia don't really work.
Troubleshooters are supposed to believe that mutants are rare, I feel like describing to the group what has happened is almost too obvious.
It may indeed be obvious. The mutant suddenly turns into the Human Torch. The troubleshooters shoot it and then pop open a Bouncy Bubble Beverage to celebrate. The clone replacement comes in and happily greets his team, and the team knows that the last clone was a Commie who infiltrated the ranks. This clone is surely not a mutant.
I think that this attitude also should lead to some initial skepticism to some of the less dramatic mutant powers, rather than an immediate execution. Which leads to…
Encourage players to come up with spurious logic to explain their mutant powers
For powers that are mental, the result can be passed back, but for describing physical powers and their eventual side effects (glowing eyes and the such), how do I keep it from being too obvious to the other players that a power has been activated?
A clever Troubleshooter will explain that he has experimental R&D contact lenses or he just has a glimmer in his eye from shooting a Commie.
Mac-R-Thur-2: Say fellow troubleshooter, I do believe that your eyes just glowed. Is there something you wish to tell us?
List-R-Ine-6: That's my experimental anti-hypnosis contact lenses that I received from R&D. It most certainly is not a mutant power.
Mac-R-Thur-2: Well it does my heart good to hear that you are not a mutant. But wait! I didn't see you get contact lenses during our trip to R&D. Are you authorized for it?
List-R-Ine-6: Of course, it was under Subsection 94b, paragraph 2 of our mission orders. You did memorize it, right? And I installed them while your back was turned at R&D so as not to distract you from inspecting your new weapon.
Mac-R-Thur-2: Flawless logic. Carry on!
I usually make it so that how much they can get away with corresponds to how many clones they've gone through and the time left in your session. If you have three hours to go and they're on clone 4, let them get away with more. If you have half an hour left and they're on clone 2, raise your standards of spurious logic.
Let them shoot the mutants when it's obvious
This is Paranoia after all. If the players don't feel like they're allowed to shoot mutants, it won't be as fun.
Mac-R-Thur-2: Citizen Syn-R-Gy-1, why is your hand on fire?!
Syn-R-Gy-1: …Uh, I'm just testing the new R&D hand warmers! [Lights fully on fire] And body warmers!
Mac-R-Thur-2: Citizen List-R-Ine, was this in the orders you memorized?
Syn-R-Gy-1: Open fire on the commie mutant traitor!
But of course, Syn-R-Gy-2 will not be a clone because Friend Computer would never make a mistake like that.