Tear it our regrade sorry no quick fox for puddles or pounding just ask your local city they sure don't fix them because they are a pain to do correctly.

HI -- Sorry , can't be fixed . You would have to replace . There is patching asphalt that comes in bags , but it usually doesn't last long . To rid the areas of water you could drill a few holes for drainage and pound some short lengths of pipe in each hole to keep them open . Or trench the bad areas and install drains with PVC pipes and grill drains .It's something to try . Maybe phone around to some paving companies to see what they say . There might be some new ways of dealing with this now .

Is this your driveway? If it is and if you have enough slope going from the lowest puddle point to the street then you can cut a 12" band in the the asphalt leading to the street and stub the drain at the street. This will take the water from the puddle to the street.