Only, she began to be afraid of the ghastly white tombstones, that peculiar loathsome whiteness of carrara marble, detestable as false teeth, which stuck up on the hillside under tevershall church, and which she saw with such grim painfulness from the park.
It isn't a clause connector, it's more of a reference on the feeling which the marble colouring brings.
That feeling you get when you hear a sound in your house in the middle of the night.
In the cited excerpt, the author adjusts the focus of the narrative in midsentence, from "the ghastly white tombstones" to "that peculiar loathsome whiteness" of the marble from which the tombstones are carved. The substance of the sentence wouldn't change if we reworded the first part of the sentence as follows:
Only, she began to be afraid of the ghastly white tombstones, and in particular of the peculiar loathsome whiteness of the carrara marble...
Presumably the author has opted to use that instead of the in the original version of the sentence in order to emphasize a familiarity with the "peculiar loathsome whiteness" that has captured the narrator's (and the character's) attention.
As Janus Bahs Jacquet's comment above explains, the that attaches forward to the later singular noun that it modifies ("whiteness") rather than backward to the earlier plural noun ("tombstones") that immediately precedes it. For this reason, that (not those) is the correct demonstrative determiner to use, given that the author had chosen to use such a determiner instead of using a simple definite article ("the").