My experience
Amps sound best into the wall
Digital components (DACs and Streamers) benefit from plugging into a Power Conditioner and also from upgraded power cords. So far not the case with my Amp.
Passive Power conditioners clean up the sound without negatively affecting the imaging, soundstage and high end frequencies as active conditioners do in my experience.
And to the OPs original question, yes the power cord for the conditioner affects the sound.
I tried a PS Audio Regenerator and a Audioquest Active Conditioner but found the best performance for my system with a Shunyata Gemini 8 passive. Also I preferred the Shunyata Gamma power cable to the more expensive Theta cable.
Even though my digital components ( Aurender and MSB)) have provided excellent power regulation they benefited from the Shunyata Conditioner.