Is it permissible to wipe over the two socks again after taking them off and putting them on anew, and after one's ablution has been invalidated, knowing that they were wiped over previously during the first ablution?

What is attributed to Sheikh Al-Islam Ibn Taymiyyah, that removing the khuff (leather socks) does not invalidate ritual purity (tahara), is correct. This is the madhhab (school of thought) of Al-Hasan and Qatadah, and it was favored by An-Nawawi, Ibn Al-Mundhir, and Sheikh Al-Uthaymeen. This opinion is strong in its evidence, and whoever acts upon it, their prayer is valid. However, it is not permissible to put the khuff back on and wipe over them after removing them, except after a complete ablution (wudu') that includes washing the feet. So, what is invalidated by removing the khuff is the wiping (mas-h), not the purity itself, because the purity was completed by the wiping according to the legal evidence and is not nullified except by legal evidence.