Hello! I've had this problem with the knife tool for 4 consecutive models, each time I encountered it I looked online and couldn't seem to find anything to help since I'm pretty much new to blender and don't know the terms. Each time I encountered it I had to trash the model, I tried again with a different blade/technique but to no avail. Whenever I cut onto a surface with the knife tool and try to move the vertices, weird lines show and connect to other vertices without any line in between them. The problem doesn't appear in object mode, only when I render it. Apologies in advance if my formatting sucks, this website is confusing me a little bit lol.
It looks like you’ve created a polygon, and are now basically bending/twisting it in someway by moving a vertex. Everything in computer graphics is secretly made up of triangles. The reason is that if you take a square and twist it in 3-D space so that the vertices are not all coplanar, you can’t really define how you’re supposed to interpret it. So the computer secretly breaks it up into triangles, and if it’s twisted, you can clearly see where it’s broken up. To avoid this arbitrary looking mess, you basically need to either make sure you move the vertex coplanar with the face it is part of, or just triangulate your mesh.