If it is a real stainless steel pot, you should be okay. Yes, they can "burn" if you let them boil dry -- been there, done that.
The question isn't a core, since it is really thin. The question is what the blend of metals it is made of. Not all "stainless steel" is the same. If it was really cheap, as in "ACME brand made in China" cheap, there is no telling what it is really made of (Just ask Wile E. Coyote).
I don't think it will hurt you, but you may never be able to get rid of that corrosion looking discoloration, which may annoy you enough to ditch it. What I can see in your photo doesn't make me worried.
IMO, if it was cheap, and you will lose sleep over it, toss it and get a new one. I have a similar pot that has a steamer insert and a pasta insert. It does what I want it to do, but if I ruined it somehow, I'm not all that attached to it.