I am not sure what clinic you were thinking of going to, but the clinics that are listed in our clinic finder as free generally require no ID which means that it does not matter what your status is in Canada. You can even be a tourist or have no status and there would be no charge.
I would give the clinic a call to book the appointment and also ask them at the same time if it is free for everyone. I am not sure what the wait time for an appointment is as every clinic is different.
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