Is praying with thinly woven socks and allowing water to reach underneath. Is it permissible to wipe over them also?

According to the majority of scholars, it is permissible to wipe over socks when they are made of leather soles, are suitable for walking, and are donned while in the state of purification. The textual basis for the permissibility of wiping over socks is the hadith reported by al-Mughirah ibn Shu’bah that “the Prophet wiped over socks and sandals” [Ahmed, Abu Dawud, and al-Tirmidhi - who declared the hadith authentic].

The majority of scholars consider that the general permissibility of wiping over socks mentioned in the hadith is restricted by the [other] hadiths concerning wiping over khuff [leather foot gear], they thus consider the conditions for wiping over socks to be the same as the conditions for khuff.

However, some scholars – like some Hanbalis, and like the recent [Jamal al-Din] al-Qasimi and Ahmed Shakir – took the apparent meaning and permit wiping over socks without restriction: whether thick or thin, and whether fully covering the feet or with holes).

Based on the above, wiping over transparent socks is forbidden according to the majority of scholars, but permissible according to a few scholars.
The above answers what is mentioned in the question.

And God the Almighty and Majestic knows best.