Is the prayer valid for someone who wipes over his socks during official working hours, knowing that he wears them after Fajr prayer until before Asr prayer, and that he sometimes washes his right foot and puts it into the sock before washing his left foot? And what is the best in this situation?

To summarize how to wipe over socks (khuffs) and its conditions, the most important condition is that they must be worn after a complete purification. If one foot is washed and inserted into the khuff before the other foot is washed, then wiping over the khuffs is not sufficient, according to the majority of scholars, if you incur impurity and wish to perform ablution again. This is the view of Ash-Shafi'i and Ishaq, and similarly from Malik. However, Abu Hanifa, Sufyan Ath-Thawri, Yahya bin Adam, Al-Muzani, and Dawud permit wearing them after an impurity and then completing purification. Therefore, if you sometimes do this, the prayer performed with this second ablution is not sufficient according to the majority, and you should make up the prayers you performed in this state. But if you wear the socks after a valid purification, it is permissible to wipe over them for one day and one night for a resident, and three days and nights for a traveler.