For jets shut down is the only option, sometimes if the jet just flames out with no associated 'bangs' or vibrations etc, then maybe go for a restart.
Not sure what to say about pistons, I had a partial failure with some vibration once but elected not to shut down and use what power it was giving to RTB asap, turned out to be a bent push rod/stuck valve ( in retrospect I would have shut down, I spent all that time doing that exercise on test and then come the day I didn't do as my training dictated.