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] (* [This long words][1] were typed by @snailboat: Only a little of which were originally posted by me, and then he added the long words to clarify my question , thankfully.)

This highlighted part, so that ~ that between complements and adjuncts, is very hard to understand. Above all, I can't guess what 'that' in the words means. Can you explain the meaning of the that and the whole meaning? [1]: When adjectives become predicatives, do they also become adverbs?

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] (* [This long words][1] were typed by @snailboat: Only a little of which were originally posted by me, and then he added the long words to clarify my question , thankfully.)

This highlighted part, so that ~ that between complements and adjuncts, is very hard to understand. Above all, I can't guess what 'that' in the words means. Can you explain the meaning of the that and the whole meaning? [1]: When adjectives become predicatives, do they also become adverbs?