The book I'm currently reading has a "C.P. - (number)" footnote every 20 pages. What does C.P. mean, and what is the purpose of this marking?
This is a signature mark.
I've often seen this in older bound hardback books, although the practice has passed out of use with more modern bookbinding techniques. Probably the copy of this book that you're reading is a few decades old.
The letters are the initials of the book's title: in your case, "C.P." stands for Consider Phlebas (I did a quick web search for one of the quotes seen in your images, to be sure what book it was). The number is a section number: you will find these footnotes running from
C.P.–1 up to at least
C.P.–8 and maybe further, depending on the size of the book. They'll only be seen in the first half of the book, of course, because the binding is such that the pages in the second half are on the same double-sheets of paper as those in the first half.
Further information on signature marks - although that page focuses on much older books, from a few centuries ago rather than just a few decades, when signature marks looked rather different and more archaic.
As Rand al'Thor says in his answer, this is indeed a signature mark. I am adding this answer to note that, if these marks are present, they will occur throughout the book, not just in the first half, at intervals of 16 or 20 pages (or some other multiple of 4 pages), almost always at the left of the bottom margin of a right-hand page.
If you look carefully at the top of the spine of a traditionally bound hardback book (it helps to open the book so the covers are at about 60 degrees to each other), you will see that it is assembled from sets of 4 or 5 sheets of paper, folded in half (so that each sheet has four pages printed on it) and stitched to the binding; these sets of pages are called "signatures". If you open the book to a page in the middle of a signature (page 103 or 123 in the case of the OP's book), you will see the stitching.
(I would make this as a comment, but I don't have the reputation.)