What is the ruling on the prayers of one who did not intend, during ablution, to wash a nail due to being unaware of its presence, and does this fall under leaving a condition of prayer out of ignorance?

One must turn away from and ignore obsessive thoughts, as this is the most beneficial cure for them. Delving deeply and meticulously into matters of purity increases these obsessive thoughts, and some scholars have considered it an innovation (bid'ah). Ablution (wudu') is valid as long as the Muslim performs it according to the prescribed legal method. A single intention (niyyah) is sufficient for all the limbs; there is no need to specify an intention for each limb. If the person performing ablution makes an intention at the beginning of the ablution, and washes their limbs as stated in the verse from Surat Al-Ma'idah: ﴿O you who have believed, when you rise to [perform] prayer, wash your faces and your forearms to the elbows and wipe over your heads and wash your feet to the ankles﴾ [Al-Ma'idah:6], then their ablution and prayer are valid.