The correct option is C She sings at the club every night to earn some extra money.
When there is more than one adverb/adverb phrase in a sentence, they are placed in the order: adverb of manner, place, frequency, time, purpose and so on.
The underlined words are: adverb phrase of frequency "every night" and adverb phrase of place "at the club". The order here is: frequency before place, whereas the correct order must be vice versa. Hence the sentence must be changed and option A is incorrect.
Option C is correct as it has the right order of adverbs- here the adverb phrase of place "at the club" comes before the adverb phrase of frequency "every night".
Option B is incorrect for the same reason as A, the adverb of frequency (every night) has been placed before the adverb of place (at the club), whereas it should be placed after.
Similarly, option D is also incorrect. (The order in these sentences are not right, for the reason mentioned in the beginning.)
Note: The adverb is usually placed directly after the object. For example: She ate her food quickly; 'quickly' is an adverb of manner.
When the object is too long, ("to earn some extra money" is an infinitive clause that performs as the object of the verb "sings") the adverb can be placed before the verb and the object.