Water that was used for the morning Netilat Yadayim is considered impure, as it contains the impurity that had descended on the hands during the night. Therefore, if this water fell on the ground, one should not step on it. Once the area is dry, there is no longer any concern, and one may walk on the ground where the water had been.
If one washed his hands in the morning into a basin, and had yet to discard the water from the basin, may a person learn or pray in the presence of this water while it is exposed?
The Hida (Rav Haim Yosef David Azulai, 1724-1806) writes that the custom among the especially pious was to refrain from learning or praying in the presence of this water, due to its status of impurity. The work Helek Levi questioned this stringency, noting that the Shulhan Aruch permits learning Torah and reciting Berachot before one washes his hands in the morning. If Torah learning is allowed before one washes, then certainly it is allowed in the presence of the water used for washing.
Hacham Ovadia Yosef explained that the Hida disagreed with the Shulhan Aruch, and followed the view of the Zohar that one may not recite Berachot before washing his hands in the morning because of the impure forces that descend on the hands during the night – which is the opinion that we follow. Quite reasonably, then, he approved of the practice to refrain from learning or praying in the presence of the water used to remove this impurity.
Hacham Ovadia further notes the ruling of the Magen Abraham (Rav Abraham Gombiner, Poland, d. 1682) allowing one to wash Mayim Aharonim (the washing at the end of a meal) over the floor under the table, and then recite Birkat Ha'mazon. Since nobody walks on the area underneath the table, there is no reason to prohibit washing over that part of the floor. The Magen Abraham clearly assumes that it is acceptable to recite Birkat Ha'mazon in the presence of the water used for Mayim Aharonim, even though that water is exposed. One might, at first glance, extend this ruling also to the case of water used for the morning Netilat Yadayim. However, Hacham Ovadia dismisses this line of reasoning, given the possibility that the impurity on the hands in the morning is treated more severely than the status of one's hands before washing Mayim Aharonim. Indeed, the impurity on one's hands in the morning is eliminated only by pouring water over the hands three times, and not just twice, clearly indicating that this impurity is more difficult to eliminate. By the same token, then, even if one may recite blessings in the presence of the water used for Mayim Aharonim, this might still be forbidden in the presence of the water used for the morning washing, given its severe level of contamination.
Indeed, Rav Haim Palachi (Turkey, 1788-1868) writes that one may, strictly speaking, recite Birkat Ha'mazon in the presence of an open bowl containing water that was used for Mayim Aharonim, but not in the presence of water that was used for the morning Netilat Yadayim. And both the Hida and the Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) were of the opinion that the water used for Netilat Yadayim has a more stringent level of impurity than even urine, such that one may not learn or pray in the presence of this water just as this is forbidden in the presence or urine.
In conclusion, then, on the level of strict Halacha it is permissible to pray or learn Torah in the presence of the water with which one had washed the morning Netilat Yadayim, though there is a basis for stringency as a measure of piety. However, Hacham Ovadia ruled that if one pours a Rebi'it (approximately 3 ounces) of water over this water, then this negates the impurity and there is then no room for stringency at all.
Summary: If water that had been used for the morning hand-washing was not discarded, and it is exposed, it is permissible to pray or learn Torah in its presence, though it is a measure of piety to refrain from doing so. One who wishes to be stringent should cover the water or pour on it a Rebi'it of ordinary water before praying or learning in its presence.