It is difficult to give you specific exercises since I do not know exactly how small your arena is, nor any details about on what you and your horse are wanting to work on at home. I believe that you will accomplish the most if you identify, in advance, precisely what you and your horse need to improve. This makes everything you do have a real purpose, and will make it much easier to have purpose toward your riding between lessons and set exercises that will help you progress.
While a small arena can get to feel very restrictive when that is the only place you have to ride, there is still quite a lot that you can do even in a limited space. In fact, some sorts of work are best accomplished in smaller arenas. For example: work on turns and angles, as well as lateral work, and increased collection in your flat work.
There are a couple of general principles to keep in mind when setting fences in a restricted space. First, remember that the shorter approaches to your lines will dictate using somewhat shorter distances between fences since your horse will not have the same open and forward stride approaching and between related jumps. Also, the jumps themselves can be lower and narrower than you are currently doing in lessons and shows, yet still be effective. The bigger questions are asked in the approach and landing rather than the jumps themselves. My second suggestion is not to try building a course or multiple lines with jumps set on the outside of the arena. Instead leave room to go around fences in order to better use diagonal lines; they will offer more room and allow approaches that can be ridden smoother. ‘Square’ corners put your horse on a very short stride and keep you from maintaining smoothness and an even pace out of the corner and between fences. Especially in a narrow arena, leaving the rail open on at least one side, if not both, will give you many more options.
My book “101 Jumping Exercises for Horse and Rider” (available from Amazon and at many tack shops) offers some specific exercises that will fit in smaller arenas, as will shortened versions of some others. Decide what it is that you want to accomplish and choose exercises that will help you toward your goals, whether it be a stronger and better balanced position for you, encouraging your horse to place the exact center of his jumping arc over the center of the jump, improving the efficiency of your turns in jumper or equitation classes, or some other aspects of your performance. Your trainer is the one to go to for help in this regard, and it is a good idea to run some specific exercises past her to see what she thinks; after you have determined that an exercise will fit nicely into your home arena.
Even if you don’t do much jumping when riding at home, placing some ground poles or cavaletti in strategic places in your arena will sharpen up your execution of courses—and will keep your at-home rides much more interesting for both you and your horse. Even in your flatwork, learn to be creative in coming up with specific “tests” for you and your horse to execute. Using all three gaits, lengthening and shortening, circles, lateral work, halts, and rein back (with transitions at specific points in the arena). There are an infinite variety of patterns to create, learn, execute, and perfect. I generally prefer a ‘three tries’ approach: get it all done the first time (even if it is a bit rough), then improve it the next time through, and make it the best you can do in your third attempt. Don’t drill after that—do something different. Endless repetition will make your horse either frustrated or dull. You will know when you and your horse are truly functioning as a team when you can execute a rather demanding pattern; perhaps with some poles or cavaletti interspersed with transitions, changes of gait, and other tests. When you can convey to your horse exactly what you want and when, on the first try, you will find you are riding a very attentive and interested horse—one that puts a great deal of trust in you as a rider that makes his job infinitely easier for him to accomplish.
Good luck and enjoy the opportunity to use your creativity as you progress toward real mastery in this great sport!