[mhio]

How would feeding air outside of an engine increase performance?

Unless the gas is accelerated inside the engine bell, I don't see a force being applied to the rocket.

[addaon]

This approach is often called a thrust augmentor, e.g. https://doi.org/10.1023/B:FLUI.0000045678.92653.98. Basically you're using excess energy in a high-speed jet exhaust to accelerate ambient air, increasing the total momentum at a lower exhaust speed.