Athjikhar should actually be athjhikhar but one of the /h/ must have been lost by GRRM somewhere. It means sickness so the city would be the "city of sickness".
What should the correct spelling of the city be then? (Vaes Athjhikhar, Vaes Athjhikhari, etc.) The printed map has Vaes Athjikhari.
It should have been
Vaes Athjhikhari. The
/-i/ at the end is optional though in this type of construction. The
/-i/ is a genitive decelnsion so that is what give the "of" in the English but with these cities the genitive ending can be left out and it would still translate the same way.
As for what is correct that is another question. The misspelling is what has been printed so that would perhaps make in canon in a way.