The Shulhan Aruch (Orah Haim 328:36) writes about a certain gum-like substance that people would chew as a remedy for dental problems, and he writes that it is forbidden to chew this substance on Shabbat. Since this is chewed as medicine, it violates the prohibition of Refu’a (healing) on Shabbat. However, the Shulhan Aruch adds, one is allowed to chew gum for the purposes of freshening his breath. Bad breath, the Shulhan Aruch writes, is not considered an "ailment" for purposes of Halacha, and so "curing" it is allowed on Shabbat.
Based on this ruling of the Shulhan Aruch, Hacham Ovadia Yosef writes that it is entirely permissible to use mouthwash or other breath-freshening substances on Shabbat. One is allowed to gargle mouthwash or spray his mouth to freshen his breath on Shabbat, as this does not constitute "Refu’a." By the same token, it would be permissible to apply non-medicated powder into one’s shoes, or onto one’s socks or feet, to eliminate the odor.
In the next passage (328:37), the Shulhan Aruch writes that any food which healthy people normally eat is allowed to be eaten for medicinal purposes on Shabbat. A classic modern-day example would be drinking tea with lemon and honey to alleviate a sore throat. Even if the patient drinks the tea solely as a remedy, to soothe his throat, it is nevertheless permissible, since tea is something which even healthy people drink. The same would apply to sucking a candy to alleviate pain in the throat.
Another example is drinking prune juice to cure constipation. Since prune juice is also commonly drunk by healthy people, and is not used only as a remedy, one may drink it as a laxative. Hacham Ovadia also mentions the example of eating vegetables as a remedy for heartburn, which is entirely permissible, since vegetables are also eaten by healthy people.
Summary: It is permissible to use mouthwash or other breath-freshening agents on Shabbat. Any food or drink which people normally eat or drink may be eaten or drunk for medicinal purposes on Shabbat. Thus, for example, it is permissible on Shabbat to drink tea to alleviate a sore throat or to drink prune juice to alleviate constipation.