I had a clicking noise on my 05 grand prix when cold....
Thought it was new engine time
Turned out to be the plastic shield under the transmission that covers the flywheel....when cold this piece of plastic was rubbing on the flywheel causing a hell of a noise....
Took it to my mechanic we removed the plastic cover cut off the part that was rubbing and no noise all winter
Hope your noise is as easy and cheap as it was for me...good luck!