Yes. One may touch these areas using a cloth.
Ketzos Hashulchan 2:2 based on 46:2; M”B 46:6; Artzos Hachaim 27
Analysis on above ruling:
The Ketzos Hashulchan rules based on Admur 46:2 that it is permitted to touch the eyes etc using a cloth. Seemingly the novelty is that the evil spirit that resides on the hands prior to washing does not affect the areas of the body if there is an interval in-between and hence only direct contact can damage. However it requires further analysis as to why at all it is permitted to touch the cloth, as Admur rules based on the Zohar that one may not touch clothing prior to washing. In truth there is no contradiction in Admur [and hence the proof of the Ketzos Hashulchan falls] as it is clear that in this Halacha of 46:2 Admur is following the Talmudic ruling which allows touching clothing prior to washing, as in that same Halacha he writes to recite Elokaiy Neshama prior to washing, which is only allowed to be done according to the Talmud as opposed to the Zohar. Thus no proof can be brought from here. However perhaps it is for this reason the Ketzos Hashulchan writes the word “Mapah” cloth as opposed to “shirt” which is the word in Admur, as even according to the Zohar the prohibition is only on clothing and not on a mere cloth [such as a napkin or handkerchief] and the proof from Admur is merely that the evil spirit does not damage through a cloth. Vetzaruch Iyun!