Is it true that utensils made of glass or clay that are broken from the corner or somewhere else must be cast off? Is there a hadith that says something like that?

All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallahu ‘alayhi wa sallam, is His slave and Messenger.

We could not find any hadeeth or a scholar's statement indicating that broken utensils made of glass or clay must be cast off. Rather, it is allowed to use them for any valid purpose. Anas, may Allah be pleased with him, said:

"While the Prophet, sallallahu ‘alayhi wa sallam, was with one of his wives, one of the Mothers of the Believers (i.e. one of his wives) sent a bowl containing food with a servant. The wife (in whose house he was sitting) struck the bowl with her hand and it broke. The Prophet, sallallahu ‘alayhi wa sallam, joined the broken pieces and put the food back in it and said (to the Companions), 'Eat.' He kept the servant and the bowl till they finished eating, then he gave another unbroken bowl to the servant and kept the broken one." [Al-Bukhari]

However, the broken spot on the utensil should be avoided so that it would not cause harm. Abu Sa‘eed Al-Khudri, may Allah be pleased with him, narrated that the Messenger of Allah, sallallahu ‘alayhi wa sallam, forbade drinking from the broken spot on a qadah (drinking bowl). [Abu Daawood and others] Shaykh ‘Abd Al-Muhsin Al-‘Abbaad mentioned a number of reasons for the forbiddance in this regard and then remarked, "It is also possible to say that the broken spot may be sharp and thus injure the person, unlike the even rim of a cup. In brief, it is forbidden to drink from the broken spot to avoid potential harm."

Allah knows best.