Question

Identify the adverbial phrase in the following sentence:

Answer 1

The only thing you need to do to identify the adverbial phrase in the sentence is to find an adverb or a characteristic which contains an adverb.  As you can see, here is  a characteristic made up by an adverb which I have highlighted:  The studious Andrea will go to the store after she finishes her homework. So, the correct answer is definitely  ''The studious Andrea''.

Answer 2

The whole question is completely wrong - there is no adverbial phrase in this sentence, since a phrase cannot contain a finite verb in itself. What you are looking for is an adverbial clause, and thus the correct answer is after she finishes her homework, because it acts as an adverb - it answers a question when something is going to happen.
The studious Andrea is a noun phrase, will go is a verb phrase, and her homework is also a noun phrase.