Dear Mr. W:

You wrote:

>In the sentence ‘they made their way slowly…’ what is the the name of the part of speech ‘way’?

It directly follows a possessive pronoun; that gives you a good clue that it is a noun. It is a noun and the direct object of the verb “made.”

You also wrote:

>I am trying to identify and name the grammatical structure of ‘must see’, ‘must have seen’, ‘to smoke’, and ‘smoking’.

The first two are verb phrases. “Must” is an idiomatic auxiliary verb like “can,” “may,” or “shall.” Some authorities consider “see” as an infinitive or “have seen” as a perfect infinitive following “must.” That is the pattern in Latin. However, most authorities simply see them as verb phases: “Must see” in the present tense and “must have seen” in the present perfect tense.

“To smoke” is the infinitive. It can refer to the act of inhaling tobacco or other substances or it can refer to a fire or other heat source producing smoke.

“Smoking” is either a gerund or a present participle, depending on how it is used in the sentence. In the sentence “he quit smoking last month,” it is a gerund because it is acting as a noun. In the sentence “We came across a smoking campfire,” it is a participle because it is acting as an adjective. It is also part of any progressive verb phrase as in “Before he quit, he was smoking two packs of cigarettes a day.”

It sounds like you are studying basic English grammar.