There is no such etnicity as Khazar. Khazar Kaganate was multiethnic state, so "Khazars" is like "Americans", purely citizensship, not ethnic name.

Mongols, like many other Middle Asia ethnicities including Chineese get some Neanderthal DNA traces only after contacts with Indo-Europeans.

May be Ainu, the indigenous inhabitants of Japan had natural Neandertal DNA traces, like European tribes, but study of real history and antropology of Ainu is kind of "cancelled" and not welcomed by modern Japan authorities.