Q. Is part of a fingernail that is loose and hanging. a Chatitzah and must be removed before washing Netilas Yadayim.

A. A fingernail that is loose and partially hanging is a chatitza and should be removed before washing netilas yodayim.
Shulchan Aruch (Y.D. 198: 21) rules in regards to mikva immersion. that a hanging nail that is mostly attached is a chatitza, but not if it is mostly separated. Shach and Taz (ibid.), and other Poskim differ on the reason. One given is that when it is mostly separated, water enters easily, unlike when the gap is small. There is also the issue of “kol haomed liktzoz” or anything that is to be cut, is in Halacha considered as already separated, even when the nail is mostly separated it would be a chatzitza on the small attached area. (See Piskei Teshuvos O.H. 161: 5, Shulchan Aruch HaRav 161:4, Kaf HaChaim 161:4 and others).
Horav Shlomo Miller’s Shlit’a opinion is that on the onset all partially separated nails should be removed. When they were not, netilas yadaim should be repeated without a brocho.

Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit’a