The last half of Mark 6:11, as found in the King James Version ("Verily I say unto you, It shall be more tolerable for Sodom and Gomorrha in the day of judgment, than for that city") is not found in any modern translation of the Bible. I have, however, been unable to find any commentary on this reading and why it's been excluded from recent translations.
Usually the NET Bible footnotes would have something on this, but there's nothing there. And even Metzger's Textual Commentary ignores it.The papyri are not extant for this portion of Mark, which is not surprising since Mark is not well attested in papyri from Egypt. So here is the state of the manuscripts with their conventional dates:
Sinaiticus and Vaticanus are both missing this passage, which is no big surprise since they tend to drop things out.
Alexandrinus has the passage but Ephraim, Bezae and the Freer manuscript do not.
The versions are split. Several important families of manuscripts do have the passage, in fact the majority of manuscripts do.
I cannot think of any controversy from the fifth century that would explain the sudden inclusion of this material.
The first half of the verse implies something like the second half of the verse, and in fact the parallel passage Matthew 10:14-15 has the passage. The second half of Mark 6:11 is Matthew 10:15 verbatim with the omission of the word land as in "for the land of Sodom and Gomorrha." Mark does tend to omit material from Matthew.
This looks like one of the better cases where material from the parallel passage has been copied over. I would judge it to be a case where Mark omitted the material from Matthew, and because the situation is implied by the wording, the copyists, starting in the fifth century, added the material from Matthew back into Mark.
So, the bottom line is that Jesus probably said it. Mark left it out of his epitome to soften the implications of rejecting the gospel, but starting in the fifth century, the copyists added it back in from the parallel in Matthew. If you want to cite it, you are safe just to cite Matthew 10:15.