Don't cut anything yet. There are some other things to consider before you do that.

It is possible that at least part of the problem lies with you, not with the kayak. Perhaps you are quite experienced in kayaks and what I am about to explain is irrelevant, but I think it is worth considering before you attack that beautiful deck.

Is your technique creating the problem? It is not unusual for beginners to use a paddling technique that causes tracking problems. If you use long, slow strokes in a kayak, that brings out the worst aspects of the kayak's handling. Trying to paddle harder just makes it worse. Your paddle stroke is off-centre, so each stroke has some turning effect. That is unavoidable, however on some kayaks it is hardly noticeable. A lot of kayaks turn slowly at first when you finish the paddle stroke and the turning becomes faster, so the longer you leave the paddle out of the water, the faster the kayak will be turning when you finally get the paddle back in the water on the other side to counteract the turning. So what you need to do is reduce the time between the end of one stroke and the start of the next.

The back part of the paddle stroke has more turning effect than the front part of the paddle stroke. As a rough guide, your hand should not come back past your hip. I have had some success helping beginners who have been going around in circles by telling them to stop doing the back part of the stroke. Just put the paddle in the water well forward, paddle gently, just do the front half of the stroke and get the other paddle blade back in the water as soon as possible. It is the last part of the stroke that makes the kayak turn. So what you are aiming for is short, gentle strokes with a high cadence.

The way to stop a kayak turning out of control is to catch it early before it turns too quickly for you to get back under control. Short, quick strokes help you do that. After a while, catching the kayak before it gets out of control becomes instinctive, a bit like riding a bike.

Even if you have paddled before, it might have been in a kayak that was designed so that you could get away with a long, slow stroke.

It is possible that you will still have to do something like add a skeg. You may even have to cut out the cockpit and build a new one with the back much closer to the stern. It isn't quite as destructive as it sounds. However, if I were you, I would be trying everything else first. It will cost you nothing to try a modified, shortened stroke to see if that makes enough difference.

I appreciate that you probably want a kayak with strong tracking rather than good turning ability, but it may be that if your technique is contributing to the problem, that a slightly less drastic modification might be enough.

On a happier note, I really like the shape of that accent area on the rear deck and I assume you have pinstripes between the strips. I can understand why cutting into that very tidy recess behind the cockpit is not something that you are looking forward to.