Many people use the expression "enhance your chance" to point out ways to increase your chances of winning or earning a contest or prize. When you enhance something, you heighten it or make it better.
"Hance" is not a word, but the addition of "en-" does something to enhance it and improve its sound. You can enhance the size of something, too, by altering it or raising it, which is what the word original meant (the "hance" part came via French from Latin altus, meaning "high"). When you enhance something you take it to a higher level, like adding salt to French fries to enhance flavor or adding words to enhance your vocabulary.