Earlier quoted context omitted.
Yes. See: https://en.wikipedia.org/wiki/SABRE_(rocket_engine) https://en.wikipedia.org/wiki/Skylon_(spacecraft) But there are lot of problems with this and it adds a huge amount of complexity.
Maybe too small an effect, but what if they put 8 NACA scoops on the sides of Falcon 9 and just fed the air in between the center and outer engines. It would get accelerated out with all the regular exhaust, and I suppose would cause a small back pressure on the center engine in particular, but they could throttle it slightly if that were a problem.
Unless the gas is accelerated inside the engine bell, I don't see a force being applied to the rocket.