I think you can do an OK job repairing it with just PVA glue. Brush it into the spine. Use a thin layer, not too much. You will want to place parchment paper between the pages at the site of the split. Then close the book, hold it for a couple minutes, press in a bit on the spine to help the cover on the spine attach to the drying glue. Place a couple rubber bands around it to hold it in place (optional), place it under a stack of other books for weight, leave it alone for a day while the glue cures.
I recommend looking up "paperback cracked spine repair" on YouTube to see examples of professionals doing the repair so you can learn. PVA glue is acid-free and dries flexible, which are the two most important qualities in book-repair glue.

