Jump lesson - making the turns

Today's lesson was not as spectacular (riding) as Saturday, which was unexpected, since I did MORE riding, but that meant it was more educational.
We started with some work on the flat (same as last lesson) because Duke was stiff and I couldn't get him round and on the bit.  When we started jumping with a right turn to a cross rail, fine, which John gradually raised, but when I turned in across the center, instead of down the longer side, it was harder to make the turn.
Then John had us do the four stride line, and after a few times through, I started getting really upset and asked him for a break to talk about it.
John said that Duke drifts left, which is just his thing - John doesn't know why, but Duke has consistently done it, and so if he's having a day where he's stiff, and that will certainly happen at some show in the future, it's going to be a harder ride (same as for dressage).
So what do I do?  I use my left leg and my left rein to keep him from going left, I ride a little bit to the right (so if he makes a jig at the last second, we're ok), and I look through the turn.  He said that I only need the left leg and rein until he moves over, and then not to use it, and that he's seen horses who then start going right, and then go in the center for a while, and then go back to drifting left.
It was a really helpful discussion; it isn't that I'm riding him differently from one day to the next, it's that he's less soft some days than others.  He also reminded me what it was like when Duke arrived, and how much he's improved since then, and how much better this is to deal with than a horse who is a nasty stopper.
So we went back to doing the 4 stride line, went down the center to a skinny, and then to the oxer, then back around to the 4 stride and Duke was much improved.
So it was a great insight, and a great learning opportunity, but not quite as pat-on-the-back as jumping that oxer at an angle to the barrel.