I've had similar problems sometimes with sink drains. I'm guessing it's leaking were the rubber washer and the sink bottom connect? If so:
Make sure the black washer is fitting correctly. It should almost fit into the opening, but still have part of the curved section showing. Tightening the plastic nut will force more of the round section further and tighter into the sink bottom.
Check the sink drain opening for any irregularities. Run a finger tip around the drain edge and feel for bumps; it should be perfectly smooth.
I'm guessing the clicking sound you heard may be a stripped thread on the drain or a cracked nut. The PVC plastic is very easily cross threaded if not lined up correctly. Sometimes you can "uncross" the threads with a sharp utility knife.
If the two suggestions I first mentioned don't apply and nothing is working it might be worth noting that the type of drain in your photo (the accordion or flexible-type) is not up to code anymore. It may be easier to purchase a new rigid P Trap.
Can you provide any pictures or a more detailed description?
I presume these are normal threads and you are trying to turn the wrench counter-clockwise to loosen, right? :-)
It sounds like there may be corrosion involved, which will make it difficult to loosen. It could also be good old-fashioned pipe-dope that is thoroughly set and hardened.
A longer handle on that wrench might help (Archimedes, levers, etc.).
Also, impact can help a lot. Tapping it with a hammer while trying to turn it might do the trick. Or pounding the handle with a hammer in a counter-clockwise direction might do it.
Once you get it to start turning, the rest should be easier.
Of course, depending on your tolerance for replacing part of your plumbing if it breaks, enough raw physical force can solve a surprising number of problems...
Best Answer
Brute force applied carefully would be the answer in this case. It's not unusual for adhesive material to seep under the drain grille when assembled. An angled pry-bar that can fit through the mesh to get as close as possible to the edge of the circle for best leverage and minimal possibility of damage.
The square holes would give you the best flexibility of attack when approached on the diagonal. The tricky part is to have something thin enough that won't bend when you try to pry.