I've read in numerous sources that intercooling improves the efficiency of a Brayton cycle. The explanation given is that it decreases the compression work by minimizing the temperature during the compression process as shown in the top picture here: https://imgur.com/a/GxSgMfL?. However, during the next step of the cycle, the isobaric heat addition, wouldn't additional heat need to be added to make up for the heat lost during compression process? To demonstrate what I mean see the Pv diagram in the bottom picture here: https://imgur.com/a/GxSgMfL?. Intercooling would result in point 2 being further to the left on the diagram, then additional heat would be required to go from the new point 2 to point 3 as compared to the heat required to go from point 2 to point 3 without intercooling, so it is unclear to me why efficiency would improve.
The intercooling increases the density of the charge so increasing the amount of air for the burning process.
While there is a reduction of heat - that small amount relative to the amount added by the combustion process, any reduction is more than covered by the improvement due to the density and subsequent charge increase.
I fitted an intercooler to a turbo charged engine - was warned by some garage mechanics that the engine would overheat and explode.
We did a whole load of measurements and theoretical analysis and fitted it anyway. Result - the oil temperature of the engine went down by some 10 or 12 degrees C - with an increase in power and efficiency. Note we were at university with access to the equipment as we were studying engineering...