What is the ruling on erring in the manner of wiping over the socks (mash al-jawrab), such as wiping over the top of the foot and a part of the heel without intending to do so? And does the sock becoming soiled with mud affect the validity of wiping over it? And is it obligatory to repeat the prayers that were performed after this wiping?

Wiping over the top of the sock, including the heel, is sufficient and does not invalidate the wiping. Wiping over the khuff (leather socks) with a thin layer of mud does not cause harm. However, if the mud layer is thick, becoming a barrier, then it is not sufficient. In the event that the 'Isha prayer performed as a shortened prayer (qasr) is invalidated, it is more cautious to repeat it as a full prayer (tammah) to avoid scholarly disagreement.