Assalamualaikum ustaz. Is the prayer of a woman not wearing socks valid if she just wears an article of long clothing covering both her feet?
Alhamdulillah, praise and gratitude to Allah SWT for His countless blessings for us all. Praise and salutations to our beloved Prophet Muhammad PBUH, his family, companions and all those who follow his footsteps until the Last Day.
We start with the hadith where Rasullullah PBUH said:
لاَ يَقْبَلُ اللَّهُ صَلاَةَ حَائِضٍ إِلاَّ بِخِمَارٍ
“The Salat (prayer) of a woman, who has reached puberty, is not accepted unless she is wearing a Khimar.” 
Imam al-San’ani explained: “Khimar is anything which covers the head and neck.” 
Imam al-Nawawi said: “Covering the aurah is the valid condition of prayer. If even one of the aurah is exposed, then his prayer is invalid, even if it is a little or a lot exposed, regardless of whether it is a man or woman (it is obligatory to cover the aurah).” 
According to the final opinion in madhhab al-Syafie, the soles of woman is considered as aurah and they are obligatory to be covered in front of non-mahram men as well as in prayer. If they aren’t covered, then his prayer is invalid, whether it is the obligatory or sunnah prayer. While, in terms of the type of clothing used, the shariah didn’t specify the type of clothing that should be used. It depends on what is easily found by the mukallaf. Moreover, in fiqh book, it is stated that if no aurah covering clothing can be found, then it is permissible to use mud in covering one’s aurah. The prayer with mud used to cover one’s aurah is valid and it isn’t necessary to replace the prayer when there is nothing else that can be used to cover one’s aurah for mud is pure and covers the aurah the same as clothing does. 
Hence, the prayer of a woman not wearing socks but long clothing that covers her soles is valid. May Allah give us good understanding to us all.
Let us all supplicate to Allah SWT:
اللَّهُمَّ أَعِنَّا عَلَى ذِكْرِكَ ، وَشُكْرِكَ ، وَحُسْنِ عِبَادَتِكَ
“O Allah, help me to remember You, to give You thanks, and to perform Your worship in the best manner.”
 Narrated by Abu Daud (641)
 See al-Mu`tamad fi al-Fiqh al-Syafie, 1/213