We are here to provide the public with a service.
In tasks candidates are presented with a point of view, argument and problem.
Sources are Oxford learner's dictionaries ; provide and the explanation of Cambridge IELTS.
As far as I knew, Adverbs of kinds are of place, time, manner, purpose, cause, reason, etc, and the adverb phrases beginning with above modifies verbs in each sentences ; Provide and Are presented.
Than, in which category are those AP included to an adverb of kind?
To get the terminology right you should not call these "adverb phrases". They are "prepositional phrases". These seem to be obligatory complements of the verb. The verb "provide" requires an indirect object in the form of a prepositional phrase
In the first example, the direct object is "the public" (and this direct object receives the thing that is provided) and the prepositional phrase gives the thing that is provided.
The second example (which is passive) could be rephrased in the active voice to simplify analysis:
In tasks, (someone) presents candidates with a point of view, argument and problem.
Again the verb "present" has a direct object "candidates" and an indirect object in the prepositional phrase which gives the things that get presented.
