We think that there is a book sounds better to me. Any other thoughts on this?
I believe that the closest translation to "there is a book" is "Є книга" which literally means that a book exists or is. I have not heard of "there" being used in Ukrainian other than pointing to a location.
Also, the order of words is important in Ukrainian depending on what we want to emphasize:
Там книга implies: If you go and look there, you will find a book.
Книга там implies: The book that you are looking for is where I am pointing to.
Please clarify if " що " refers to what we are thinking, or to which particular book we are referring to. If the latter is accurate, " We think that book is there " should also be accepted. (It wasn't!). However, these two uses of " that " have distinctly different meanings, and it would be helpful understand precisely how these distinctions are expressed in Ukrainian.
" що " refers to what we are thinking: it introduces the (what we are thinking) clause. " " що " is mandatory even when in English it is omitted or understood. To express which one: "that" = той, та, то depending on the gender of what we are describing. "this" = цей, ця, це.
So if we wanted to say "We think (that) that book is there", we would say
Ми думаємо, що та книга там.