This Halacha addresses the amount of water required to be “provided for the hands,” meaning the act of one person pouring water from a vessel over the hands of another, who is holding them out to receive the water for ritual purification. It was uncommon in those days for a person to wash their own hands, as that required more water.
Rabbi Yose holds that the final person washing must receive at least a quarter-Log of water, implying each pair of hands requires no less than a quarter-Log. Thus, he disputes the view that a smaller amount suffices for a pair, rejecting the entire first half of this Halacha.
The final part of this Halacha distinguishes the second washing from the first, which is for Terumah. The first washing requires a single pour to cover the entire hand, preventing the water from becoming impure before the hand is fully purified. The second washing, to remove residual impure water from the first washing, allows water to be "added," meaning poured incrementally over parts of the hand to clear remaining drops.
The final part of the Halacha’s structure, addressing the second washing before the first, indicates this Halacha's focus is on the second washing. This explains why less than a quarter-Log may suffice, as the second washing requires only enough water to remove residual drops, unlike the first washing’s need for a full quarter-Log. Rabbi Yose, however, maintains that both washings require a full quarter-Log.