The correct options are
A She lived in the middle of the great woods.
B The desert stretched towards the north.
An adverb phrase is a group of words that does the work of an adverb, that is, modify the verb.
Options A and B are correct. Both sentences contain an adverb phrase each, which modifies their respective verbs. In A, the phrase "in the middle of the great woods" tells us where the subject "she" lived. That is, it modifies the verb "lived", and is thus an adverb phrase. In B, the phrase "towards the north" tells us how far the desert stretched. that is, it modifies the verb "stretched". It thus is an adverb phrase.
Option C is incorrect as the sentence contains an adjective phrase. Such a phrase does the work of an adjective. that is, modifies the noun. The phrase "enormous wealth, but little patience" can be rewritten as "wealthy, but impatient". It modifies the noun "Tashi", and it thus an adjective phrase, not an adverb phrase.
Option D is incorrect as it does not contain an adverb phrase. "To go home" is the object of the sentence; it does not modify the verb "wants". Thus, it is not an adverb phrase. It is a noun phrase.