Is one allowed to was their hands with soap after washing for bread? (i.e. wash for hamotzie, then use some soap and then dry your hands). The purpose of this is too avoid as many germs as possible-protecting you from the germs on the washing cup as well.
In principle, it is fine to wash one’s hands with soap after netilas yadayim, and this will not somehow “cancel” the netilah. However, this is not proper practice, because of the interruption between the act of washing and the blessing of “al netilas yadayim” that is made after one finishes.
Rather, it is better to wash one’s hands with soap before doing netilas yadayim, and only then washing properly for bread with a vessel. I one is concerned for the germs on the vessel, I suppose one could wash for bread, recite the berachah, and wash one’s hands again before eating.
Although this is also not strictly lechatchilah (because of the interruption), it can be considered as being “for the purpose of the meal” (if you are concerned about the germs on your hands), and therefore permitted.