Do the threads from socks stuck to toenails constitute a barrier preventing water from reaching them, thereby invalidating wudu' (ablution), and must the prayers performed with this wudu' be repeated?

It appears the questioner is afflicted with waswas (obsessive doubts), and they are advised to completely disregard it. As for their question, if it is merely doubt and waswas, then nothing is entailed. However, if it is something certain and the barrier was slight, then some scholars have made concessions regarding it, stating that it does not harm. This is the choice of Sheikh Al-Islam Ibn Taymiyyah, like slight dirt under the fingernails or in the nose or cracks. It is permissible to adopt this view also due to the matter having passed. As for the view that ablution is invalid even if the barrier is slight, this is the position of the majority of scholars, and according to this, the prayers would be invalid. For someone who does not suffer from waswas and wishes to be cautious in their religion, they may repeat the prayers they performed.