that will depend on how much is left of the cord on your Bose. you could go ahead and cut the damaged part of the cord and remove the isolation. Then do the same for the part with the plug. Remove the insulation from the cable ends by about 1/4 to 1/2 length. You could now use some wire nuts, crimp connectors or just tist each of the wires together (there should only be two wires consisting of a bunch of single stranded copper wires) Once twisted together (or otherwise attached like crimp connectors/wire nuts/soldered) wrap some electrical tape around the repaired area or use some heat shrink to cover it.
If you do not have enough on your radio to that repair, you will have to disassemble it and desolder the damaged wire. Then solder a new power cable to that. You can use one of the inexpensive extension cords you can get the department stores and cut it to length. Let us know if you need more help with that or if anything is unclear. you can always post some good pictures with your QUESTION so we can see what you see and help you further if needed.
Here is a good video about how to do it. You do not need all those fancy tools…..;-)
“Buy a whole new radio?” Absolutely not!