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There isn't too much you can do to restore wicker if it is totally "over the hill" and broken, aside from actually reweaving it with new wicker... a real skill! But if it is merely dingy looking and dirty, you can give it many more years of life.
Briefly, wicker care is a two-fold process... proper cleaning followed by a finish coat of protectant. Take the piece outside (oops... it is outside, isn't it?). Wash it thoroughly with a detergent such as TSP or TSP-substitute to remove all dust and dirt. If you think there is mildew on the wicker add a cup of household bleach to each gallon of cleaning solution. Mildew usually appears as small black dots or discoloration that may not wash off.
Let the detergent/bleach mixture remain on the chair for at least 5 minutes before rinsing thoroughly with clear water. If you have already washed the piece and find lingering mildew, you can spray it with a 50/50 bleach water mix and rinse after 5 minutes or when mildew disappears. Bathroom mildew removing sprays work well too, but be sure to rinse thoroughly with fresh water!
The wicker should dry for at least a day before applying a finish. Water that becomes trapped in the weaves can take a long time to evaporate.
Any clear finish such as varnish, shellac or lacquer can be used to keep that "natural" look. Flat finishes will give the most natural look... glossier finishes will add more durability and cleanability. Exterior polyurethane works great, too! You can also paint the piece if desired. If the piece is always outside, be sure to only use products designed to take the wrath of Mother Nature.
For ease of application, the clear sealers can be sprayed on with an inexpensive plastic spray bottle, available at most hardware stores. After 10 minutes, wipe off excess sealer to prevent the formation of "tacky" spots.
Paint can be either sprayed or brushed. Canned spray paints work particularly well, but expect to use quite a few cans!