What is the ruling on a small part of the neck and hair of women becoming visible during prayer, and is that small amount which is difficult to guard against excused, given that it appears and is covered repeatedly during prayer?

If there is doubt about whether a part of the body became uncovered during prayer, the default assumption is that it did not become uncovered. If it is a result of waswasah (whispers/obsessive thoughts), then it should be ignored. If a small part of the awrah (private parts) became uncovered and the worshiper promptly covered it, then their prayer is valid by consensus. If a small part of the awrah became uncovered and was not covered, the majority of scholars hold the prayer to be valid, and this is the view preferred by Shaykh al-Islam Ibn Taymiyyah. It is recommended to secure one's clothes well to avoid the awrah becoming uncovered, and not to pay attention to waswasah related to this. If the worshiper promptly covers whatever part of their awrah appeared, their prayer is valid. Even if they did not promptly cover it or did not know until the prayer concluded, their prayer is valid according to the majority of scholars.