It depends on how broken the disk is. If the problem is in some files, most programs will be able to burn there. If the problem, however, is the TOC (table of contents), I don't think it can be done. And even if it gets done, I seriously doubt the mess will be possible to read.
Even if it's the first case, I seriously advise against such an attempt - if some of the files broke, close to the middle, the ones closer to the periphery are in even greater danger. Discs get scratched easier towards the periphery.