Hi Blastframe,
I have done a complete series here on Cineversity about Blend-Modes, if you like to dive into them.
In short, the Add mode summarize the values that you have in the background and foreground image. In this way you get the pixels values from both added. There is only one result possible, brighter than the anything in the scene, except for absolute black areas. Anything else—again brighter.
The simplest way would be darken (blend-mode) the top layer. Simple means here that you get not the right saturation or color in the shadows. As you have here a dark street, you might not be worried about that too much. All these blend modes work here only to a certain degree, which means, not really my suggestion in most cases; if we talk about photorealistic approaches. For Motion Graphics, in an artificial environment, that is a complete different story and much (again) simpler.
Multiply leaves everything that is black in the foreground as it is in the background, but any light in the foreground layer will “alter” the result.
If you work in integer (not float) some people like to use a mid gray as base “color”, and then use just Overlay in the compositing. I think for anything photorealistic we should use a linear workflow with practical footage from a calibrated and profiled camera, as well any image used in C4D only from profiled and calibrated cameras as well. BTW, to save an image in a floating point format does NOT linearize it. If the creator of the image has done adjustments to it, it might be ruined anyway. Many texture and HDRI supplier “fix” their stuff with some adjustments. A reason why I always suggest to use a MacBeth card. I discuss this all in my next tutorial series about texture, hdri and such.
I use the shadow pass normally as mask in an adjustment layer. With that I can control the shadow much more sophisticated. This allows to take the existent shadows in the background image as reference and adjust brightness as well as saturation/color, perhaps even the contrast, separately. If you measure the existing (background shadow) with the eyedropper, you get the RGB value for this, and a good idea how to adjust your shadow accordingly.
There are other ways to prepare the scene in C4D already, by using the practical footage, but you asked for an compositing fix, but I thought it is worth mentioning.
All the best
Sassi