What is the technical name for this kind of phrase?

Solution 1:

Since the phrase is headed by with, uncontroversially a preposition, the whole phrase falls into the category of prepositional phrase. That is its form.

As for the function in clause structure, it is best analyzed an adjunct of reason. That is to say it is a grammatically non-essential part of the clause which adds optional information - in this case an explanation of why the situation expressed by the rest of the clause is true.

Refer to the Cambridge Grammar of the English Language p732

(c) Implicated reason

There are also, as so often, various constructions which have a more general circumstantial meaning, but which may be interpreted in appropriate contexts as giving a reason for the matrix situation:

[27] i Having known the candidate for ten years, I can vouch for his reliability.

ii With six people away sick, we can’t meet the deadline.

It is not an appositive phrase as those are almost always noun phrases where one may substitute for the other, which is clearly not possible in this case. Refer to the Oxford Dictionary of English Grammar p31

Apposition A relationship of two (or more) units, especially noun phrases, such that the two units are normally grammatically parallel, and have the same referent, e.g.

Our longest reigning monarch, Queen Victoria, reigned from 1837 to 1901

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