There's a lot training can do.. but training can't fix pain (though rehabilitative training, especially force free, can improve it, and forced incorrect training can make it worse). Forceful training can temporarily suppress the symptoms of pain, if the horse fears the consequences enough, but you'll always be in a battle against the pain... with the horse caught in between.
By trying to train pain you're putting your horse between a rock and a hard place. Do they respond to their body telling them to not do something because it's causing damage? Or do they respond to the human, who's demands are becoming increasingly more frightening or uncomfortable, and risk causing more damage to their body?
Some common pain symptoms that we try and train might be - rearing, bucking, bolting, nervousness, fearful reactivity, biting, kicking, yanking legs away, sitting back, resistance to the bit/bitless, hard mouth, reluctance to bend/flex, hollow backs, dropping rails, refusing fences, girthiness, saddle sour, barn sour, herd bound, moving at the mounting block, reluctance to move forward or stop under saddle... I mean.. I could literally go on forever. These all sound like common "training problems" don't they?
So how do we know when we are trying to train pain or if it really is something that is a "training issue"?
Unfortunately we can never guarantee a horse is not in pain. Even if you've had a dozen vets and all the therapists in the world look at your horse, you can not look at that horse and say they are for sure NOT in pain. Nobody can say that, just like nobody can look at you and tell you that you are NOT in pain when you clearly feel pain. So what this leaves us with (I'm my experience and opinion) is being as compassionate and understanding as possible towards our horses in training and in care.
We can choose to avoid escalating pressure and punishment, to reduce the risk of putting our horses between a rock and a hard place.
We can choose to give our horses the benefit of the doubt when an issue shows up, rather than automatically assuming the worst of the horse; that they were trying to take advantage of us or being a brat that day.
We can choose to give our horses more choice and control in the training, giving them the opportunity to choose to participate or not. Accepting that when they choose not to participate that something is "off" (training wise or physically) or they aren't feeling well enough.
We can choose to treat our horses like living beings and not sports equipment or machines without feelings and emotions or aches and pains.
We can choose to educate ourselves constantly on how better to care for our horses and prepare them physically for the tasks we ask of them.
We can choose to put aside our biased opinions and traditions and opt to try something new and different, if we think it might be in the best interest of the horse.
We can choose to advocate for our horse and never rest on the opinions of just one(or many) professionals or friends.
We can choose to not settle for what's easiest or convenient for us, and do what's in the best interest of the horse... Even if that means more work or changing our goals and expectations.
The same thing goes for nutritional and environmental causes of behavior we often try and "train away". We often try and fix with training what is being caused by high sugar feeds, artificial feeding schedules, or lacking nutrients in the diet. Sometimes it works temporarily, but it won't last.
Training also can't fix what's lacking in the horse's environment and social life; nervous energy from confinement, anxiety from frequent transportation and moving, panic from isolation.. all of these can be temporarily suppressed but not "repaired" by training.
We may never be able to fully say that our horses are without pain, and we may not be able to offer them the same life and diet as a feral horse (but with the added benefits of man made comfort and health care), but we should be able to say we've done our absolute best. That we haven't forced our horses to work through their pain, and with an inadequate lifestyle or care, for our pleasure. That we haven't tried to train away things that had nothing to do with training, just because it was easier for us than changing the pasture or diet or hauling the horse to the osteopath.
I'm not saying we will never make mistakes, trying to train something that was actually pain related, but we can try and do better. We can also choose training paths that reduce the risk of forcing a horse through discomfort and distress. By being patient and giving our horses a choice we allow them the ability to communicate to us that something is wrong, as long as we are willing to listen.