pre vs amp power cable question

The actual requirement of the gauge of copper in a power cord, (at least in Audio) has no relationship to what cords are said to be great, vs trash.
Thei discussion has been had hundreds of times here and on many other Audio forums. And never fails to turn into a bashing 'drunken' slugfest over the 'real' truth about wires.
I doubt it could ever be settled between the wire is wire folks, and all wires are different folks.
So good luck thinking you will actually GET an answer..
Though, in general, bigger IS better in audio power cords...

In short, the answer to your question is "yes."

Of course the "necessary" gauge of the power cord is related to the required current draw, that’s why 30 amp circuits require a larger minimum sized wire (10awg) than 15 amp circuits (14awg).

My amps each have 8A fuses and I run them off individual 20A circuits. I use power cords made with 11awg wire and they sound great.

The other stuff is subjective and you can read here for hours about gauge, silver vs. copper, stranded vs. solid, shielded vs. unshielded, geometry, EMI, RFI, ferrites, connectors, conditioners, etc., etc.

My experience is that power cords do affect the sound of components but how each manufacturer accomplishes that, and whether the effect is positive or negative is still subjective to the listener and their specific system. There are a few power cord manufacturers whose products have a somewhat "universal" appeal and you can read about them on these forums. LessLoss and Isoclean are two that come to mind. Oyaide connectors are well regarded (but not necessarily their cords), and also Furutech connectors. At the lower to mid price range, Elizabeth would say Pangea Audio offers good sounding cords (I have never heard them) and others would say try the DIY offerings from VH Audio. My best advice is to either use appropriate sized generic cords and don't worry about it, or to listen to a variety of cords (borrow and trade from your friends or dealer) and decide for yourself what you like.

The easy answer is the more current required, the lower the gauge wire is needed. It is that simple. With that being said, it does not hurt you to use a low gauge power cord on your preamp. IMHO, it will be difficult for you to tell the difference sonically between properly sized power cords as the power supply inside of each of your components has more more influence over the sound of your component than a power cord. This is especially true if your device has a switching power supply where the power is regererated at a higher frequency.

If the wire in your power cord is thicker than the wire that connects the wallplug to the electric power company, you have to ask yourself why...