Naked Baby LOVES apple cores. Loves apple, period.
What is important to remember is
gagging is normal, beneficial, and protective. If baby is gagging, just watch them. Odds are
extremely good they'll work it out on their own, and that trying to help by sticking your fingers in there will only make things more likely to go wrong (hey, just like birth!

).
Having food in their mouths even after nursing is pretty normal, although other than tiny pieces I don't think NB has ever done it, but I've heard from lots of moms whose babies do. If they don't want it in there, they'll thrust it out, or gag it up if it's too far back.
The first time Naked Baby had meat, it was at Thanksgiving, not too long after he started experimenting with solids. He absolutely INSISTED I give him my turkey thigh bone. There was still some meat on it, which he loved sucking on. He eventually got a pretty big piece off, and he decided he didn't want it, and it was too far back in his mouth, so he spent about 30 seconds gagging it out. It was a pretty big piece, so it was some pretty impressive gagging. A small part of me was running around in circles in the back of my head yelling and screaming and imagining the worst, but I just sat and watched him, and he got it out eventually, and then went back to gnawing on his turkey bone.
Gagging is normal, beneficial, and protective.. Scary, too. But normal.
As for whether you should do anything differently in the future about what you give him, that's entirely up to you and your comfort level. Can you watch him gag, calmly giving him the space he needs to work it out on his own? Or is it going to be way too stressful for you, or are you going to want to interfere?
There aren't right answers. I do believe it's best if we can let babies experiment with food and their bodies, including practicing gagging if they need to, but
there aren't right answers. It's up to you, and your comfort level. You don't have to feel completely confident and calm, but it really helps if you can project that to your baby, to tell them you trust them. If you're not there,
that's fine. You don't have to give them an apple core just to prove anything.
OK, now I'm just babbling. I hope some of that helped.

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