In such systems, the proper matching of the prime mover output with the requirement in steam energy is the key to high efficiency.

If the prime mover (engine) is running at the same output (power) whilst the exhaust temperature is fluctuating between higher and lower temperature, this will mean that the Steam generator output is fluctuating (The demand for steam power is fluctuating). Therefore, the unabsorbed heat from the exhaust will make the exhaust temperature to rise, and when the steam demand increases, then the exhaust temperature will decrease.

Since the HRSG is the item contributing to the increased efficiency of the combined cycle as a unit, you should now make your conclusions to justify what you said in your question.