How would exhaust heat recovery into the combustion chamber affect the efficiency of a jet engine?
Extracting heat from the exhaust will cool and compress it, slowing it down and reducing thrust. Recirculating the heat back into or downstream of combustion will raise the temperature at that point and on downstream to the extraction point. The inevitable thermodynamic losses mean that the returned heat does not quite compensate for the heat drawn out and overall efficiency will fall a little.

What does improve efficiency is to draw through additional air mass in order to increase the exhaust mass flow, even at the expense of lower overall velocity and temperature. This is what the bypass turbofan does.

A more radical modification is to swap the combustion chamber for a heat exchanger and heat the gas via hot fluid from an external source; the hotter the fluid the better. I have seen this proposed for the hydrogen turbopump in some versions at least of the SABRE air-breathing rocket engine. It was also proposed for nuclear-powered bombers in the 1950s, though I cannot recall if those included turbojets as well as ramjets.