If one makes a mistake in a review, will the admins let you correct it after it is approved? If so, how? The editing function seems to disappear.

if they agree, they will put the review on hold again where you can make the edit and then republish it.

I've done this a couple of times where I described some activities that the provider wished to remain unknown

You are given some time to edit the review, but I believe it expires when the review is posted, or shortly thereafter.  You do have to convince admin that the correction is needed, so be sure to present a good reason for making a change.

LLAP,
Swim