According to the authors of the book, adjuncts are divided into two kinds: modifiers, which are thoroughly integrated into the syntactic structure of clauses, and supplements, which are much more loosely connected to the sentences in which they’re found. In other words, adjunct is an umbrella term to cover modifiers in the NP, VP, AdjP, AdvP, and PP construction, and supplements.
Here is what confuses me: when the authors explain the main dependents in the VP construction, they make a distinction between complements and adjuncts (p.65).
The dependents of the predicator in the VP are of two main kinds: complements and adjuncts.
However, when they explain the main dependants of the other constructions (NP (p.83), AdjP (p.118), and AdvP (p.124), they somehow specifically makes a distinction between complements and modifiers, not between complements and adjuncts.
Dependents in the structure of the NP are of three main types: determiners, complements, and modifiers. (p.83)
An AdjP consists of an adjective as head, alone or accompanied by one or more dependents. The dependents may be complements, licensed by the head, or modifiers, less restricted in their occurrence. (p.118)
The structure of AdvPs is similar to that of AdjPs, but somewhat simpler. Dependents can again be divided into complements and modifiers. (p.124)
So I'm wondering why VP gets such different treatment? Is there a reason they can't specifically pinpoint whether an adjunct in VP is either modifier or supplement?
Edit1: Here is the examples of the adjunct in SIEG 2nd edition, which indicate that 'adjunct' is used to cover modifiers in the NP, VP, AdjP, AdvP and PP construction, and supplements; not just modifiers in VP, Clauses and supplements as defined in CGEL, SIEG 1
This implies 'adjunct' includes modifier in AdjP (ii), AdvP (iii), and PP (iv) (p. 214)
The adjunct in 4-iv is clearly a modifier in the NP [some item in the containing clause] (p. 224)
Edit2: It turns out the above examples are an oversight on the authors' part. They promise to fix it in the next reprint. Please refer to @BillJ comment under this question.