A variety of solutions is a noun phrase.
You know it is a phrase (or a clause), because it is a group of words with a single meaning.
(A 'variety of solutions' is one thing).
You know it is a phrase (not a clause), because it has no verb.
You know it is a noun, because you can substitute it for a noun in a test sentence.
The doctor chose an apple.
The doctor chose his hat.
The doctor chose a variety of solutions.
A phrase is never a part of speech, only a word.
Prepositional phrase
The phrase "in addition" is a prepositional phrase in which "in" is the preposition and "addition" is its object. This phrase, as a phrase, is not a part of speech, although it may function as one, probably an adjective or adverb.
A prepositional phrase.
On the mountain top is a preposition phrase, not a part of speech.
That I am is a phrase, the individual words in the phrase are parts of speech. That -- demonstrative, determiner I -- pronoun am -- be verb
adverb phrase
verb
A prepositional phrase.
A prepositional phrase.
A phrase like 'under a vow' is not a part of speech. It is a phrase that is made up of parts of speech. under is a preposition a is an article vow is a noun
The phrase "are required" is a verb in its passive voice.