I used to use a cloth between the hand and negel vasser cup when washing because I saw others (chassidim) doing it in Monsey, but I stopped because I've never seen a reason for it in halacha. I'll admit my studies aren't perfect or anything, and there's more than likely sources I haven't seen that speak about this. I'm curious what the source is for this practice, along with reasons or suggestions of what to learn to understand it.
Rabbi Kaganoff discusses the subject in This Is the Way We Wash Our Hands and during that posting gives several reasons that would lead one to use a cloth when holding the cup.
Rabbi Kaganoff explains that there is a machlokes as to whether the hands must be completely dry at the beginning of the washing and if the cup itself must be dry as well. That is why some people will hold the cup with a towel or carefully dry the hand that is about to be washed. Perhaps the chassidim that you speak of follow the psak of the Chazon Ish that Rabbi Kaganoff cites.
Washing wet hands
Must one’s hands be completely dry before you begin washing netilas yadayim? The authorities dispute what the halachah is in this case.
As we learned above, someone who, when pouring water for the first time, rinsed only part of his hand, must dry his hand thoroughly and begin the procedure over. The authorities dispute whether one must always have dry hands when beginning netilas yadayim or whether one may perform netilas yadayim even though his hands are wet or the handle of the cup is wet. According to the Magen Avraham (162:10) and the Mishnah Berurah (162:27), one may begin washing netilas yadayim, even though one’s hands are wet. The Chazon Ish (Orach Chayim 24:20) disagrees, contending that one’s hands must be dry when one begins washing netilas yadayim. Therefore, the handle of the cup must also be dry or, alternatively, one may grip the handle of the cup with a towel or some other item that keeps his hands dry until he washes netilas yadayim.
Another possibility is that the water on the cup may be considered used water, which can cause problems. Note that if the first hand is held properly, the water from the first hand being washed will not flow onto the cup so there would be no problem of tum'ah.