Optional depictive predicatives as adjuncts
Obligatory predicatives are clearly complements, dependent on the occurrence of an appropriate verb. With optional ones, however, there are grounds for saying that while the resultatives are complements, the depictives are adjuncts. Resultatives are either obligatory (as in the He talked himself hoarse construction) or else need to be licensed by the verb. Optional depictives, however, are less restricted in their occurrence. One manifestation of this has just been noted: they can occur in transitive clauses with either S or O as predicand. Another is that they can occur in combination with an obligatory predicative or in the ditransitive construction:29 i) They look even more fantastic naked. ii) They served us our coffee black.
We will therefore regard such predicatives as adjuncts, so that the predicative/non-predicative contrast cuts across that between complements and adjuncts.
Like numerous other kinds of adjunct, predicatives may be integrated into the structure as modifiers, or detached, as supplements:30 i) They left empty-handed. [modifier] ii) Angry at this deception, Kim stormed out of the room. [supplement]
The supplements are positionally mobile and are set apart prosodically. The modifiers are of course more like the complements, especially in cases where they occur very frequently with a particular verb, as with leave in [i], die in He died young, bear in the passive He was born rich, and so on.
I don’t find this modifier explanation except on this part. So it’s not clear to me whether the book says empty-handed is the predicative for they, and this enters not into CGEL’s but into the traditional category of adverbs as a modifier. When I say ‘it sounds clear’, this adjective clear is both a predicatve (for it) and an adverb (for sounds) is what the book says?
This is the answer that I've got from Linguistics Beta which is written by Tim Osborne:
The adjectives in question do indeed behave in a unique way. They are dependents of the verb, but they are predications over the subject (or object), e.g
He died young.
The adjective young is a direct dependent of the verb died, but it is a predication over the subject he, i.e. it assigns the property of youngness to he. In other words, young is behaving like an adverb/adjunct syntactically because it is a direct dependent of the lexical verb died, but it is definitely an adjective insofar as it looks like an adjective and is assigning a property to a (pro)noun.
As Jlawler points out, the terminology used to denote these words varies. Some call them depictive adjuncts or participant-oriented adjuncts. Depictive adjuncts are stage-level predicates; the property that they assign is not an intrinsic characteristic of the noun, but rather its applicability is transient. Adjectives that assign intrinsic properties cannot occur in this use, e.g.
*Bill died interesting.
Unlike young, which is a transient state because we all get older, interesting is intrinsic and inalienable. The distinction between stage-level and individual-level predicates is discussed at the bottom of the article here: https://en.wikipedia.org/wiki/Predicate_%28grammar%29.
The direct answer to the question is therefore as follows: predicative adjuncts modify the verb if one interprets modify to mean 'be syntactically dependent on', or they modify the subject or object if one interprets modify to mean 'assign a property to'. Note that typical attributive adjectives unify both of these meanings of modify, e.g.
the young man
In this case, young is both syntactically dependent on man and it is assigning a property to man.