Is it permissible for a private tutor to take payment for lessons that the student's mother canceled, knowing that the tutor might have refused other lessons for the sake of these particular sessions?

If you have agreed with the student's mother on a specific number of teaching hours per month, then you are entitled to the agreed-upon wage as soon as you are ready to teach during these hours. The student's mother's apology does not forfeit your right to the wage, because a private employee is entitled to their wage if they make themselves available for work, even if the employer does not utilize them. This is because the ijarah (leasing/hiring contract) here is based on confining oneself to the work during its designated time. A private employee is one who works for a specific person temporarily, with a fixed-term contract, and is entitled to the wage by making themselves available during this period.