Okay, so I think maybe you aren't super clear on how the clicker training works.
The whole point of clicker training is that
A) you can mark an exact movement, the very moment the horse(or whatever animal you are training) does it. It's much more instantaneous to use a sound, to reward the horse for making the movement you want, than it is to pet them or whatever, and you can make that sound without having to move, possible changing your body position and thus the message you were communicating to the horse with your body (ie- you have to get close enough to be able to pet them, which could be seen by them as you stepping toward them and they might think they are then supposed to step away). Especially if there's some distance between you and the animal, like if you're working a horse 'at liberty' or you are teaching a cowdog and he's on the other side of the herd from you, then it's really not feasible for you to get close enough to pet them, fast enough
B) you get the benefits of rewarding with treats, without actually having to give out treats.
So you can do this with any auditory cue, be it the clicker or a word or sound you make.
Before using the clicker (or whatever sound) for training, you have to build a strong association between the sound and good feelings in the animal (food obviously gives the strongest good feeling), otherwise, it's just another sound to them and it has no meaning.
So to start, you would just stand there with your horse, not asking her to do anything, and just make the sound then immediately give a piece of treat or food. Make the sound, give treat, make sound give treat. Over and over and over and over again. After a while, the animal learns to associate the sound with getting a treat. So when they hear the sound, they get a rush of 'feel good' same as they would when they get a treat. So the sound becomes the treat/reward. You are then able to use the sound, in training to 'mark' and reward a desirable behavior without having to actually give them a treat. Which is why I think it seems like it would be most beneficial with at liberty training, where you really need a lot of motivation to get a horse to want to do something, since you have no way to physically control him and no real easy way to apply pressure for a correction either, since he could just leave. You need him to be highly motivated to do what you ask, so you've got to find a way to really reward the movements you want and since you are too far away to reward with a treat or petting in a timely manner, the clicker training solves that problem.
I can also see its application in regular horse handling as well, since it allows you to reward every correct behavior with a 'treat' (the clicker, which the horse has been trained to associate with a treat), without having to carry a billion treats with you all the time. Plus, it would give you a way to do it while riding as well, since you definitely can't be carrying a bunch of treats and feeding them to your horse while you're riding (also why I think it would be better to use a word like 'good' instead of the clicker, since you can't carry and use the clicker when you're riding but you can certainly use your voice).
Though I can see where it could be handy, I don't really think it is THAT beneficial to regular horse handling to have them clicker trained, since most horses are highly driven by release. I've really never had trouble teaching horses things using only release as the reward. Horses, being prey animals, are a lot different in that respect than dogs, who are predators. Being let out of work/pressure and left alone means a lot to a horse, but not so much to a dog. Dogs usually need food or toy rewards to really learn, otherwise most are just not super interested. Horses, due to their nature, are generally a lot more cooperative.
But, I do use these concepts to an extent in my horse training. Like I mentioned, I'll rub/scratch on a horse (usually on their neck near their withers, where I'd pet them if I was on them) and say 'good boy/girl', right after I put hay in their feeder, as they grab the first few bites(especially new ones that have just come in and ones that seem to not care for people or being petted). That way they learn to have good associations with me petting them and telling them 'good', so when I want to reward them with a scratch and a 'good', they understand that it is a reward. Otherwise the words mean nothing to them and some horses don't really care for petting (in all reality it is pressure we're putting on them), so they need to be conditioned to like it.
I don't really know how long it would take to get those associations with a clicker or other sound, since I don't specifically set out to do that. I'll just pet on them for a few seconds every time I feed them. You can usually tell when the horse starts enjoying being petted (if you can find their itchy spots, you'll have them immediately LOL).
You'd need to reinforce the idea regularly so they didn't forget that treats often follow the sound and it lose its value to them.
If I were going to pursue clicker training. I'd probably just work on the clicker-food association (clicking immediately followed by treating, without actually trying to train), until I could tell that the horse expected food to follow the sound (again, no idea how many times it would take for the horse to get it, probably depends on the horse and how motivated by the treat they are). Then, after that was accomplished, I'd probably bring a few carrots or other treats with me each day, and briefly review the clicker-food association with them after catching them, before working with them, so it's fresh on their mind, and then again when done working with them, before putting them up. During the actual time I was working with them, I wouldn't use treats, I would just use the click or whatever sound to mark the good behaviors, still using other rewards like releasing them at the right time and petting them. The click or other sound would just help get the reward timing right and allow you to reinforce the reward with a treat (well the idea of a treat), without actually having to give a treat.
From your post about tonight:
Now when I think of it, perhaps I should have given the carrot first before praise and scratching and rubbing her? The time window between when I clicked to when she got the carrot was maybe 2-3secs.
Too much going on to get the timing right here. The clicker is supposed to help clear that up, not further complicate it. That's why you need to get the positive association with the sound established first, so that when you're actually training her, you only need to worry about the timing of the sound. You can click and give release at the same time, and if it was something really difficult for her, immediately follow with lots of praise.
You need to be very aware of what movement you were marking though. Like for instance, from your post about tonight
Just worked on leading her away from me on both sides of her. 3 or 4 times on each side and figure I would do full circles, after we made a full circle each time I used my clicker and praised her "good girl, good job etc" rubs, scratched her and gave her a carrot.
Don't wait to 'click' and reward until you've made a full circle, mark each correct step. Because when you step to her, you aren't trying to get her to spin around in a circle, you're wanting her to take a step away for every step you take to her, so that's the movement you need to mark with the reward (click or sound). Marking each correct movement will make a lot more sense to her, same as you need to correct each incorrect movement as it happens instead of waiting for a string of incorrect actions to happen before you make a correction. I wouldn't mark the movement unless she did it without a fight, since that's what I'm after. Does all that make sense?