Both are correct. It depends to what the speaker intends "which" to refer. "Which" can refer to the thing (singular) that they gain, or it can refer to "experience and skills" (plural), two things that enhance their chances. Both interpretations make good sense. However, there are circumstances where only one interpretation will work. Here are some examples: "Billy finished all of his chocolate and vanilla ice cream, which pleases his mother" (Only singular makes sense because it is not the two kinds of ice cream that please his mother) "Billy read 'Hamlet" and 'Othello', which are famous plays by Shakespeare" (Only plural makes sense because it is the plays, not the reading, to which "which" refers. However, if you write "Billy read 'Hamlet' and 'Othello', which astonishes his teacher" then only singular makes sense.

The second sentence is correct, with the verb, “enhances.” The subject of the “which” clause sentence is the concept described in the first part of the sentence, and a concept is singular, like “it,” which therefore takes the “s.” I hope this is clear!