If you insert something between hot side and air to extract energy, there will be more heat resistance and lower heat transfer, but this resistance will be bigger than what you get from your energy extracting device.

If you use that new device as a cooling enchancer, you will have to provide some power. If your first cooler user more efficient method, you should just make bigger unit using more efficient method, because otherwise whole apparatus will be less efficient.

Thermoelectric elements add resistance. You are literally adding a giant resister between the hot and cold side, which decreases the thermal Flux for a given temperature difference.