Is it permissible to give monetary alms, which were designated for various purposes such as thanking Allah or seeking healing, to students with the aim of achieving positive effects on them, knowing that this benefits the teacher, or must it be spent on those who have no relation to them?

Voluntary charity (Sadaqah (definition) Tatawwo') given as an expression of gratitude to Allah and seeking healing is a voluntary act. If you are an agent in delivering it, then either the donors specify the recipients (such as the poor), in which case it is impermissible to disburse it to others; or the prevailing custom dictates its disbursement to the poor and needy, in which case it is impermissible to disburse it to wealthy students, because "what is known by custom is like what is stipulated verbally." However, if the donors do not specify the recipients and custom does not dictate its allocation to a specific group, then there is no harm in giving it to students, even if they are wealthy, because voluntary charity can be given to the wealthy. In this case, it is preferable to seek permission from the donors.