How to repair a pothole on the edge of an asphalt driveway

First of all, what you describe sounds like job done improperly. Perhaps you can even use the contractor warranty and have it fixed for free.

Asphalt pieces put from different loads will usually have perfect adhesion to each other - the previously put asphalt is still very clean when the newer asphalt is put, so new one will attach to it perfectly as if it was a single piece with almost no joints. Since you see asphalt peeling off there's likely a problem - most likely the asphalt or the base underneath it got dirty and that prevented good adhesion. No good adhesion - prepare for more peeling off as time progresses, especially since you have temperature variations and freezing water.

IMO it's worth cutting asphalt at the flaw spot deep enough and sealing it with a cold patch to prevent further destruction. If destruction progresses along the joint the same should be done along the whole joint.

You have divots in a concrete driveway.

I'd saw around the divot area, break it all out, and pour concrete, rather than anything to do with asphalt. Drill some holes in the edges and insert steel to tie the patch to the slab.

If following the "quick fix" fill-on-top approach, possibly thinset (tile cement) but trying to do anything with concrete in the winter is fraught with difficulty - you have to heat it up and keep it warm - if it freezes before it cures, it's toast.

My gut says that your quick fix with hot tar is going to be good for 6 months, at most. Additionally, it will complicate doing a good concrete repair later.