Repair AC adapter at plastic head
Best bet would be to get a replacement adapter. Often you can find a replacement at a place like a Goodwill outlet store. At the place near me they have 100's of adapters for around a dollar apiece. 

Next have to ask what happened to this adapter. How did the wire get like this? There is a distinct possibility that there was a short between the two bare copper areas and this may have burned out the innards of the adapter. Sometimes there is an internal fusible link wire that blows open due to a short and if this is the case then this unit is toast and will need to be thrown away. 

If a replacement is not to be found then repair may be possible with care. It is delicate work but if you are handy with this kind of thing here is what to do.

0) Cut the damaged wire off of the adapter near the end of the strain relief. if at all possible note which of the two wires joined to what side of the wire that is down in the strain relief. Make sure to mark the mating wires so it is clear which is which.

1) Carefully cut away most of the length of the molded strain relief that is sticking out of the adapter. Use extreme care not to cut the two wires and their insulation as the strain relief is cut away.

2) The two wires that are exposed in #1 above will be joined together side by side via a rib in their insulation. Slit the rib between the two wires so that each one becomes a separate branch. From the picture it looks like these would end up being a little over a half inch long. 

3) If it was not possible to determine which wire connected to which at the beginning then now is the time to look at the two wires on the other part of the cord. One should be identifiable from the other in some way. One may have a line printed on the wire insulation or there may be an identifying ridge molded in the insulation along the length. Look for matching markings on the stub wires coming out of the adapter. Mark with colored permanent marker which wire is which. (Note that if this adapter is one that puts out a low AC voltage as opposed to a DC voltage then there may be no identifying marks and  wire matching is not important. You can see the output spec on the adapter label). 

4) Strip about half the length of insulation off of each of the two wire stubs from #2 above. Use extreme care to not nick or cut any of the fine copper wire strands.

5) Cut the end of the remaining cord back an inch to get rid of the burned and melted insulation.

6) Separate the two wires on the cord into individual conductors by slitting the rib in between the two. It will be necessary to separate these back about 1.5 inches or so. 

7) Strip the wire ends of the stubs on the cord end back about a quarter inch or so. 

8) Cut one piece of heat shrink tubing about two inches long that fits over the outer insulation of the two wire cord. Slide this up on the cord from the stripped wire end about a foot from the end. 

9) Cut two more smaller diameter heat shrink tubing pieces about 3/4 inch long that just fit over the individual wire stubs on the end of the cord. Slide these well up on each conductor away from the stripped end. 

10) Now individually twist the stripped ends of the mating stub wire ends together in a manner that they are twisted in line and not out to the side. Solder each twisted wire pair to complete and secure the joint. Two things to watch out for here....do not over heat the wires because the heat shrink tubing that you slid up on the cord end stub could start to shrink and you want to avoid that at all costs....and be careful to not blob lots of solder in this process. Ideally the two solder joints will be neat and the completed joint will not be much larger than the wire insulation diameter.

11) Slide the two smaller diameter heat shrink tubes down the wires so that they cover over solder joint and overlap the insulation on the stub wire from the adapter side. Use a heat gun to shrink the tubes over the wire joints. DO NOT TRY TO USE A MATCH OR LIGHTER to shrink the tubing. Use hot air so that you do not damage the wire insulation.

12) Lastly slide the larger diameter heat shrink tube down from the cord and over the other part of the joint to cover over the whole thing. Once again use a heat gun to shrink the tubing in place. 

You can now test the adapter to see if it works correctly. This is best done using a multi-meter to read the voltage at the end of the cord when the adapter is plugged in. If it is a DC output adapter make sure the polarity is correct on the two contacts at the end of the cord. If all is fine you should be able to use the adapter for its normal duty. 

NOTE NOTE NOTE

This repair will leave the wire exit at the adapter very susceptible to damage if the wires are yanked on or moved around a lot. So it is suggested to wrap the good part of the cord out from the repair site along the side of the adapter and then use two long zip ties that fit all the way around the adapter and secure the cord in place at the side. This will keep strain off the repair. 

NOTE NOTE NOTE ABOVE