I have 5x42 walk on the back of my home that joins our drive way that joins the house on the end 24 wide both are joined like an L. the concrete was replaced several years ago but has sunk several inches allowing water to pool at the house. Can asphalt be poured over the old sunk concrete and continue up the entire drive to form one solid slap of asphalt. My worry is the rise and fall of winter. Winters are not as harsh as they once were . A lot of people are pouring slabs now for garages and seem to be happy.

Putting additional weight on an improperly formed slab will cause the slab to sink more.You need to improve footing and then you can place asphalt.

Comparing your situation to highway construction, hot mix asphalt works very well on a concrete base but joints and/or cracks eventually reflect up through the asphalt surface but that is not usually a major problem other than lessening the life of the walk..
However, as ma0641 has stated the unstable concrete sections need to be improved first. You need to remove the sunken concrete, and if you find evidence of water (maybe because the sidewalk slopes toward the house) either provide an outlet for the water and/or place a varying thickness of hot mix asphalt to make the new asphalt surface slope away from the house lengthwise and side to side.
Depending on the condition of the total concrete walk and how much repair of the concrete might be needed, you may want to consider removing all the concrete and placing a new concrete walk (after ensuring a good base and drainage) because the hotmix asphalt will most likely be all handwork and possibly cost as much or more than concrete.