Woah, hold up, I got this sentence on mobile with a "buils the sentence word by word" format and I had to write "The white cup is not here." But at the top of this discussion it says "There is no white cup here." Everything I have learned in Russian so far has told me that these must be different constructions with different word order, but now it says either is correct?
I think that "the white cup is not here" would be "белой чашки тут нет" and "there is no white cup here" is "Тут нет белой чашки", as the word order in Russian plays a role in the meaning. First part of the sentence contains the info you already have, last part is the unknown/new info.