Carp are super finicky in general, but the canal is even trickier. First, no matter where you are, you're generally very unlikely to catch a carp that isn't feeding or trying to feed. So when you see one in the canal just resting and hanging out under a tree, you can try casting to it, but 99% of the time, it won't care. Cruising fish have a higher likelihood of being a willing player, but still depends on the "way" it is cruising. With a feeding fish, if you're generally matching what it's feeding on, you have a great chance. Flies do matter - you need to match the hatch like trout. Not exactly the same as matching the size and color of a mayfly, but close. In the canal, more often than not, they're going to be feeding on nymphs.
At this point, I've fished the canal enough, I know which fish are worth casting to and which aren't. You'll start figuring that out once you get an idea of what the successful targets look like. I generally look for two types of fish - "bank diggers" which are fish picking food literally on the bank right next to where you're walking. You basically dab your nymph for those, but you need to be super stealthy. The other is the general feeding fish. The ideal situation is a school of multiple fish feeding together. Your fly becomes a competition then. But more often, you'll only find a single fish. Knowing the water clarity is going to be garbage, I try to focus on the sections that are only a couple feet deep. Approach them like bonefish. When you see those mud trails you mentioned (I call it smoke), I spend time watching it, even if I can't see the fish right then. So just stare. Watch where the new smoke arises and where it's fading out. See if the fish is moving in a general direction or just hovering in the area. Most importantly, watch for a flash of the tail so you know where the head is and which way it's pointed. In these situations, your sight fishing ability will be very limited, but you can have success with a strike indicator. It's still tricky getting the length and location right, but if they're feeding aggressively, they're more likely to move greater distances.
I've caught the carp in the canal in all weather during all seasons. But, I do find that on the super hot days, it will slow a little bit while the sun is overhead. In that case, focus nearby some of the little feeders into the canal, and you might have a little better success, but options are pretty limited.