I think it's probably worth adding onto this a little bit here because saying "stop playing Hanon" isn't helpful to the actual problem. (Unless you're following Hanon by the book)
I think the main problem with these kinds of studies is that, when come with instructions, is much open to interpretation. It also does not help build a better "technique" so to speak, it merely strengthens one that is either good or either bad. Now in your case it looks like it's bad.
I have read through the studies and see one (of many) phrases that causes major concern "Lift the fingers high and with precision". Now I read this as, simply ensure your fingers come off the keys rather than slur when picking up speed to create a clear sound.
I certainly wouldn't raise my fingers as high as possible whilst putting the others down because A - you will peak very quickly, it simply does not work and B - it's unhealthy you are working 2 levels of muscles here at the same time and your body doesn't like it. pulling the fingers up whilst pushing the fingers down - does that even make sense?
Next I see you respond about "stretching" have you ever "pulled a muscle" if you push muscles further than what they want to go - they break. sometimes a little break, sometimes you're not playing the piano for 6 months. If you ever have to stretch any fingers, go back and review your fingering (assuming you can comfortably reach an octave) I can just about reach a 10th and I would never stretch my 3/4 or 4/5 more than a 3rd apart that is C-E, D-F, E-G and so on.
I think you can work any fingering to prevent stretches, ignoring chords(while sometimes unorthodox, if you have smaller fingers, make the fingering pattern work for you, not for what's written)
Next you have to understand that if your goal is to "strengthen fingers" what makes fingers? they don't have muscles, the statement itself is flawed. Understanding the full motion that makes a finger (or thumb) depress a key and releases makes you realise that you can never make your fingers equal by themselves you can practice all day long but you'll never trill your 4/5 like your 1/3,2/3 etc.
Now i've said all that (and I can discuss further if required) It's impossible to tell what's wrong with how you play without seeing how you play. Either post a video, or watch your own video, try and see how your left hand differs from your right, whether it's a wrist bend, whether it's curled fingers whether it's a high arm, pain in the wrist doesn't necessarily mean the wrist is at problem, it could mean something else is at problem putting pressure on the wrist so really look at your whole body, both sides should be equal. Also really locate the problem, is it actually the wrist, or is it the lower palm, or maybe the bottom of your forearm, are there any actions of playing that cause particular soreness, identify and see how you would play in your right hand.
I will certainly try my best to help you fix this problem if I can see how you play. Lastly, going back to my original statement, why I don't think saying "quit" Hanon is useful. Because personally, putting the silly instructions aside, I have come to like the studies. I don't get aching from them and they have improved some dexterity. I think the best benefit from them, however would be playing in different keys (Rather than just C Major) because there are so many other keys, that if the exercises are played well and in different keys, I think they can lay some good foundation in practicing pieces, in the same way that we practice scales. When you learn the correct motions that are required with different studies, you can understand how to apply them when practising pieces.
Hope that helps. I want to on the whole say I agree with everything everybody else has said, because the studies rely on a lot of (for the want of a better word) ignorance here in that you can't go in a small cave for 6 months, learn these studies to 108 and come out the next Horowitz, it just doesn't work like that.