With respect to the many post on respect in this context I am going to offer an alternate explanation for the behaviour your observing in your mare.
Horse behaviour, like the behaviour of the vast majority of animals, even humans, is driven by what results in a good consequence or a bad consequence. Any behaviour that results in a consequence the animal values will be repeated or reinforced. Behaviour that results in a consequence the animal doesn't like or value won't be repeated or will be weakened. This is called the law of effect and it underpins all horse training and horse-human interactions.
In the case of your mare, the food is a consequence which she values very highly. It is known as a primary reinforcer and she along with all horses and indeed all animals, has evolved to seek and consume it. Any behaviour that she performs that gets her to food, gives her time to eat it, or prevents other horses (or humans) from accessing her food will be remembered and repeated any time she either is trying to get or keep others away from her food. Undoubtedly, when given a choice between food and doing what you ask, the food is going to be much more reinforcing than you, the rewards of sticking with the food much greater and so she is going to be much more motivated to do what she can to keep access to the food. Particularly as the things you may want her to do aren't actually that fun or rewarding for her- doing circles in a round pen, going for a ride etc.
It is highly unlikely horses have the cognitive skills to plan into the future- after all they don't need to strategise to eat grass, all they need to do is find it. Consequently, in any given moment they are behaving in ways that they believe best meets their needs at that time. Any tme she has access to that food, that's her best interest met. In order to change that you do need to expose her to a stimulus that is stronger than the reward of the food, and as many people have suggested, that stimulus may have to be quite strong to overcome the attraction of the food.
In effect, you need to tip the balance from food being the most rewarding to doing what you want as the most rewarding in that instance. To do that you will need to apply a pressure so that avoiding the pressure is a greater reward than continuing to eat. Avoidance of pressure is another primary reinforcer for horses and they, along with most herbivores have evolved to avoid it. We utilise the desire of horses to avoid pressure on their bodies as the basis for just about everything we do with them. We apply a pressure to motivate them to change what they are doing, they respond as we want, we release the pressure. The release of the pressure is rewarding, the horse is likely to respond in the same way next time it feels that pressure. From a survival perspective, when given the chance between satisfying the need to eat and the need to flee a predator who is trying to eat the horse, the horse will choose fleeing (pressure release) over the eating. That's what we tap into when we use pressure on the horse's nose, head etc to pull it away from a feed bin.
A word of caution though, in effect all horse training relies on horses seeking to escape pressure, so we need to make sure that when they do respond to our pressure cue as we want them to we release the pressure and thus reward the correct response. If we don't release the pressure the horse is likely to try other ways of getting that release and we don't generally appeciate those other ways as they can involve bucking, shying, kicking etc. However, if we release the pressure before the horse has responded correctly then what ever it was doing just before the pressure goes away is the behaviour that gets rewarded. Your mare kicking out at you is a classic example of this- she kicks out, you back away, she gets released from your presence, the kicking out behaviour is reinforced.
Lastly, methods of training which involve chasing the horse or eliciting a flight response, or making him do endless circles don't teach the horse to respect you, and there is the strong possibility that the horse will form an association between you and fleeing, which is the opposite of what is intended. Flight responses in horses are usually related to fear, and there is now a lot of research suggesting that fearful memories are encorded differently to regular memories. Fearful memories can be recalled more quickly and powerfully than regular memories and the physiological responses engendered by the fear (adrenaline, increased heart rate, increased blood flow to skeletal muscles etc) switch on very quickly. We see this when a horse shies. So we should be very cautious about frightening our horse to get it to do something in case it learns to associate us with fear and then has that memory triggered in another context and when we least expect it. This is called spontaneous recovery and probably accounts for the majority of "out of the blue" hyperactive behaviours we experience with our horses when they buck, shy, rear etc for no apparent reason.
Rather than chasing your horse to catch it, using food (a primary reinforcer) is often highly effective, especially if you give it the food and then walk away. (Can be hard when there are several horses in the field). Advance retreat is also very effective, it is simply a form of pressure release which rewards the horse for standing still by the person backing away a few steps each time the horse stands still. It does take patience and the ability to read the body language of the horse so you back off before the horse starts to move away. I have used it successfully but am too impatient these days and use food and can call my herd of up to 20 horses and they will reliably canter up to the gate for their treat. We have big paddocks 60+ acres each so this is very handy.