Do I need 10 gauge power cord if I have 10 gauge from panel?

NO! Because even a 14 gauge power cord is rated to handle 15 amps/ 1500 watts. This is the same rating for the typical breaker box/AC outlet found in most homes. That's a lot of power! Increasing the size/diameter of a power cord is only required if you plan to use an arc welder! Then you need a lower gauge/thicker cord. For a 200wpc amp a 14 gauge cord is perfectly fine. You don't need a garden-hose size power cord!

I would not worry so much about the gauge as the intent of the cord and the quality (it is likely to be 10 gauge) These days most cable companies build a special cord for amps as opposed to low power components.

I searched for nearly a year to get the power cord right on my amp. Finally I found the Audio Quest Hurricane. Wow, this is a really good power cord… the dealer who brought it over told me his employees asked for it because of all the buzz on line… the buzz. There is good reason for all the buzz. It is very neutral and quiet.

I guess the short answer is, yes definitely you want to find a better cord. 

it is unlikely a 14 gauge power cord is not negatively impacting the sound.

No.

I am unaware of any amp requiring a 10 gauge power cord. What amp/amps do you have? Why did you install 10 gauge wire to the breaker box/AC outlet? None of your gear combined will draw that much current. Stoves, refrigerators and washing machines/dryers run on 240 volts/30 amp circuits.