Probes to the outer Solar system use a Jupiter flyby for gravity assist, all six such missions thus far have. While flying near a gravity well one can enjoy the Oberth effect which multiplies the acceleration per unit propellant consumed.
Have any spacecrafts using a Jupiter gravity assist fired their (small) rockets in order to profit from the Oberth effect?
What does the trade off look like between using all fuel at Earth to get into the Jupiter trajectory, versus saving the upper stage and much of its fuel for a big Oberth bonus blast at perijove? New Horizons passed by Jupiter only 13 months after launch, and took another 101 months to reach Pluto. Would a slower fuel laden first leg of the trip with a big burn at Jupiter have shortened its total travel time to Pluto?