Young's double slit experiment with a single photon

There is no way to measure the photon without disturbing it. So the version of the double-slit experiment that you describe above has to be done with an electron, which you can measure in flight. It's typically called the "which way" experiment.

In your description above, the interference pattern disappears. What you are describing is a variant of the quantum eraser experiment. I have not been pleased with the descriptions and explanations of the experiment that I have found, but I'm told it makes sense if you look at the math.

A common misunderstanding is that the wavefunction describes a particle (e.g., a photon or an electron), and that interference is something that happens to a particle.

Neither is true. In fact, the square of wavefunction at a point describes probability of detection of the particle at that point. Interference is evidence that the "probability of detection" for all points follows a wave equation. That implies that interference can only be detected by performing a large number of identical detections, and not by performing a single detection.