Does Power Cord Require Burn-In To Sound Good?

I installed a Nordost QX4 a few months ago and it has made a significant and positive difference in my system. More soundstage and more depth. 
Power and its handling is the heart of the system. Power cords and a conditioner. Try Nordost. 

I assume you have a power conditioner of some type.  If not I would suggest a Shunyata with their cords which have been broken before shipping.   Use NR cables for equipment and XC cable to connect from the wall outlet to the power conditioner.   Also, replace your wall outlet if you have not done so with a hospital-grade outlet.   The outlet will provide better wall connectivity. 

in my experience, the power chord is the least effective piece of gear to affect sound quality in my rig.

The question is do you hear SQ improvement on other types of cables after break in? If you do, then of course AC cords also need break in. If you don’t, my next question is do you hear SQ improvement after components break in? 
My personal experience with AC cords is thick cables need more time than thin cables. And if you move them around, same rule apply with lesser time.

But I know some do not hear any difference. First you need to be very familiar with the sound of your system. 2nd is analyzing SQ changes  is nothing like listening to music. It does take time to learn what to listen to.

But again, I know non believers will not hear anything different still.

This is a great article - I suggest everyone reads it

https://www.gcaudio.com/tips-tricks/why-power-cables-make-a-difference/