LenaR

Not sure why would you want to send your Ukrainian passport anywhere...
 
If you mean a renounce of  your Ukr. citizenship by that, then it's not how it works.  It's long and exhausting process  in Ukraine and  turning in your passport doesn't make the process complete.  I know just one person. ONE, who went through the entire process and he says if he knew how bad it would be, he wouldn't even tried. 
  
USA doesn't care, as long as you leave and enter the country as US citizen. 
 
As of now Ukraine doesn't recognize a dual citizenship, but there is no "punishment" for having more than one either.
 
Now, Parliament is working on recognizing multiple citizenship since apparently a half of the country (at least)  possesses more than one anyway. 

I keep mine. And use it when enter/exit Ukraine . Was actually quite  beneficial during Covid lockdown when "foreigners" weren't allowed in. 
 

You have to check with Ukraine embassy about it. Some countries allow dual citizenship, some don't.

Scan everything.  You never know whether you will need them or not.

 

I've always joked with my wife that when she is finally done with her naturalization, we are going to have one of the biggest bonfires and BBQ in our city courtesy of her immigration documents.