Answer:
B. quite.
Step-by-step explanation:
Adverbs are those words that signify or give an added attribute or modify the adjectives or verbs of the subjects. They express a relation of place, time, manner of action and even circumstance or degree of the verbs.
In the sentence,
"The mystery writer has won quite a few awards for his work"
we have the subject / noun "Writer". The verb in the sentence is "won", which will make the word 'quite' an adverb. The word 'quite' emphasizes the nature of the action 'won', thus modifying the manner of winning.