I went through this with our warehouse. We got some tar repair whose name I no longer recall but it was a silvery colour with black streaks when you stirred it. Medium thin stuff so it soaks in fairly well.
Go to a good roofing supply company - don't use the cheap stuff from the retailers.
The old patching you have looks terrible from point of view of leaking. You should try to remove as much of it as you can, including that felt. The felt looks as though it wasn't completely soaked through with tar. The repair stuff may leak through the opening around the pipe so be carefull - maybe get some new felt and soak it well and then cram it around the pipe so the liquid won't run along the pipe. Use a lot of the repair stuff in several layers. Takes about 5 or 10 minutes between layers.
Your problem occurred because the tar dried out and cracked either from shrinkage or from being too stiff to flex as needed. You can keep the new repair from drying out by sticking on a layer of cloth or plastic that is UV resistant. The roofing supply company should have something you could use. In a pinch a piece of the same roofing sheet that is on the roof could do but it would be too stiff to work with easily. Something more flexible would follow the surface of the repair and keep the sun off.
Now that I think of it - get the diameter of the pipe and ask the roofer for a collar with a skirt that would be clamped around the pipe and tarred down around the perimiter. That would look professional.