tutor or private tutor

Definitely a tutor. Private if he or she is employed by the child's family.

Thanks, centrox.

'Private' in this context is the opposite of 'public' - a private tutor (in addition to the teaching provided by the student's school or college), a private security guard (who is not a publicly employed police officer), a private road (on private land, not owned or maintained by the municipality or county etc), a private phone line between my house and my neighbour's, a private doctor (who is paid a fee by the patient) etc. My house is private, not public, property.

Thanks again, centrox.

In this case, I would say that “private” means a one-on-one tutoring session with the individual student.

I can engage a private tutor for all three of my children. I would arrange it privately (not via the school or college). It will be a private arrangement.