ref – https://dictionary.cambridge.org/grammar/british-grammar/adverb-phrases

An Adverb Phrase consists of an adverb (The adverb is the head of the phrase) and can appear alone or it can be modified by other words.

Adverb phrases modifies verbs

We use adverb phrases most commonly to modify verbs. In the examples the adverb phrases are in bold. The verbs that they modify are underlined:

I [exercise] very regularly
I [eat] quite healthily
Children [grow] up.

Adverb phrases modifies ‘am/is/are’

We use adverb phrases with be. This is especially typical of adverbs of place:

I [am] upstairs. // upstairs modify ‘am’

It is (linking verb) only(adv)a spider. // only modifies ‘is’

I am (linking verb) too(adv) smart.

She is (linking verb) very (adv) silly.

Adverb phrases modifies adjectives/adverbs

We use adverb phrases (adv) to modify adjectives and other adverbs:

I found it extremely (adv) difficult (adj) to talk to her.

He drives really (adv) carefully (adv).

Adverb phrases modify noun/prepositional phrases

We use adverb phrases (adv) to:

1) modify noun phrases (np)

quite – to the utmost or most absolute extent or degree; absolutely; completely

That is quite (adv) [a tree] (noun phrase). (it’s a tree that is special in some way, e.g. it’s very big)
There is even (adv) [a place for shoes] (noun phrase).
There is hardly/barely (adv) any food. (noun phrase).
I peer over (adv) [the cliff] (noun phrase).
She lean over (adv) [the railing] (noun phrase).

Yes, adverbs can modify prepositional phrases. An adverb can provide additional information about the relationship expressed by the preposition and its object. For example:

She ran quickly up the stairs.
The cat jumped high onto the table.
He spoke softly with the visiting professor.
In these sentences, the adverbs “quickly”, “high”, and “softly” modify the prepositional phrases “up the stairs”, “onto the table”, and “with the visiting professor” respectively, providing more detail about the manner or degree of the action expressed in the prepositional phrase.

2) prepositional phrases:

I (s)
peer (v)
over (adv)
[at the funny old man.] (prepositional phrase)

I (subj)
run (verb)
daily (adv)
[at the park.] (prepositional phrase)

We (subj)
climb (verb)
right (adv)
[over the sofa.] (prepositional phrase)

Our budget (subj)
runs (verb)
well (adv)
[into six digits] (prepositional phrase).

Adverb phrases modify determiners

We use adverb phrases to modify determiners, especially words like:
– all
– some
– half
– many (quantifiers):

Only(adv) half(determiner) of my friends could come to my party.
Only(adv) some(determiner) of the survivors reached the shelter.

Very (adv) few (determiner) people have heard of my city.
[Nearly (adv) all (determiner) of the cookies] = subject
have been (present perfect verb)
eaten.