I agree with Ether. That soldering job is pretty rough looking, and yes, in competition, I would expect to see smoke emitting from it at some point.
Invest in some soldering flux, leaded solder (60/40 with rosin core is my favorite), and some heat shrink.
Remember to put the heat shrink on the wire before you solder it, and make sure it is away from the soldering point while soldering, because it will shrink down with very little heat. I know this seems like common sense, but I can't count how many times I've done it, and I will guarantee that I will do it again in the future.
Leaded solder, of course, has lead in it. No, you won't get any diseases from using it, but you should thoroughly wash your hands with soap and hot water,
ESPECIALLY before eating.
Here's how to properly solder, or at least my preferred way, and I'm pretty good if I do say so myself.
Apply a small dab of flux to the terminal and wire after you feed it through the potentiometer holes (again, don't forget the heat shrink!!!). If you get too much flux on, that's fine. The more, the better... err... you won't hurt anything by having too much.
Apply a small bit of solder to the iron. This will be used JUST to help aid in heat transfer.
Touch the iron to the terminal and allow it to heat for a few seconds (2 or 3 seconds should suffice). Solder won't bond unless both the wire and terminal are heated up enough. You'll notice the solder you applied to the iron 'flow' to the terminal once it's hot enough.
At this point, feed some more solder in to make a nice, smooth joint.
Inspect the solder joint for any bubbles, cracks, or anything out of the ordinary.
Allow the joints to cool, and then slide the heat shrink over the terminals to insulate them.
Use a heat gun to shrink the heat shrink. If you don't have one, you can use a flame of some sort, but be quick and spread the heat. Heat shrink, like anything else, will catch on fire if it gets hot enough.
If you don't have heat shrink, then insulate it with electrical tape. Just insulate it somehow. Signal wires are a magnet for shorting.