If this is something I caught during the run-up, especially at a remote airport, I would probably still fly back to my home airport. As soon as I returned, I would leave a note in the log record, and notify the flight school and and the aircraft mechanic immediately, preferably by phone. If for some reason you needed to shut off the engine during the flight, you could still do so using the mixture cut-off.

I usually don’t like giving blanket statements like the one above because every situation is different and there is not one right answer to this question. This is one of those questions that falls under the gray shade of aviation (not a black and white answer) so take that with a heavy dose of salt.

Paul.