Its something that happens in every sport. Generally target shooting the mantra is: bad day, drop everything for awhile, go get a beer, chitchat, help others in the club or simply call it the day. The worst thing one can do is fussing on a bad shot, and after that it becomes fussing about fussing on the bad shot and it never ends.
You can train yourself in many ways, meditation etc, and at the end clearing your mind is an exercise you would have to do whenever something is required from you, that is a deadline, a job interview, school tests etc.
When it comes to tournaments though, you don't have the luxury of dropping the whole thing and you must enter "the zone" from the beginning.
Eventually after a certain number of tournaments you will get the whole situation as an habit but you need to recognize the signs of things going the wrong direction and correct them before they precipitate into a point of no return state of mind.
If you can try Kyudo, the shooting there is really structured and the mental aspect of it is taken care of as part of the training as much as the technique as its considered part of it (shooting technique is much different but movements aside the philosophy behind it is transferable). What can happen messing up the concentration is that you have a lack of focus, you worry about your equipment, feel uneasy about the competition and are agitated in a general sense, at the opposite there is such a thing as being too focused, the score becomes the all important thing, you almost want to drive the arrow to the target by pure mind power, your shooting will suffer because to mental tension follows body tension, your movement will be less fluid and your technique less clean, your result poor. Then there are the others, your team, your club, your buddies, a bunch of strangers, people taking videos of the shot maybe and if you screw up there will be forever a record of it. Yet, you probably dont practice alone, your friends, team, club are always around you, videos should exist because they are a good way to review the shots.
Keep your equipment always top notch, be anal about it, even if they will be Sunday shots for the rest of your life your equipment should always be perfect and ready, all in a proper carrying bag, in order, no chance to forget at home or misplace anything ever. At competition time you wont have to worry about it, you wont have to fuss around your gear and that will keep you less agitated.
When you practice often you are in a certain state of mind, just peaceful and having fun, you can recognize the feeling and bring your mind on that, every time it happens during practice, and tie that condition to certain things to act as trigger, it can be some sequence of things you do to prepare the bow, something in your equipment, a smell... pick something constant that you wont have to prepare and fuss about before a competition, that will simply be always there and close to you (don't pick the target as those change from place to place and time). In competition you will use that trigger to ground yourself. You also might find helpful to find a calm corner to take position and draw your bow a few times without arrow (don't dry fire it obviously), if you do that do it at every practice session also.
Work out a simple routine and stick with that, practice or competition, same routine... routine is good, its muscle memory for your mind.
Breathing: that is important, your movements, since you move into position to the follow-trough, and your breathing must be coordinated with eachother, keep a slow steady rhythm, deep breaths, breathe with your belly and fill up all your lungs (in our daily life we tend to breathe faster and shallower breaths, it becomes an habit and breathing rhythm is linked to our fight/flight reflex) keep the pace the competition calls for but maintain the coordination and the depth. Deep rhythmical breathing is also a good technique to use while getting grounded (do it as an habit during practice also so its part of your routine). Some will simplify the issue just saying to hold your breath here etc but there is a significant reflex system that is worth leveraging.
If ultimately you consider the reasons that brought you to sign up for the tournament you might find things to work on: was it for fun? buddies pushed you to it? was it to show off? bragging rights? prize? Remember that in a competition you can only do your best, at that point what is there is there, the drive to improve yourself is only for practice. There should be more pressure on yourself during practice because you should try to get better and better results session after session. But it ends there, you don't get better than what you are in a tournament, the competition is just for doing what you do, it just happens that people attach points to things but really, how does it change your life? Its not a million dollars prize isn't it... just do your thing.
That others will be looking and judging your shooting its a fact, but others look at you and judge you everyday of your life and its not that big of a deal, for sure you did plenty of things way more important than shooting at a target while people judged you and didn't even think about it for a moment. Once the time comes its you and you alone taking the shot, noone else has any importance when it comes to that arrow (unless they are standing in the way ;) ).
Beside all the mental practice that should go on side by side your technique practice there are also little workarounds that are different from person to person, you will have to work them out on your own.
If you are in a team you can remove the tournament atmosphere just turning it into a bit of a game with eachother: best score pays dinner (or beer, depending on finances)... dont do worst score or you just swapped one competition for another. That way its your game that you are playing and the competition just happens to fall around it.
Back in the day one of my buddies used absurd hats while shooting, his cure for being in the centre of the attention was just to attract more attention on something that wasn't his arrow.
At the end of the day shooting is an art, its like painting on the target, and there is a certain harmony in all the little things that come together in the perfect shot, once you see it its instinct to tune into it.