der Kern
pit
The inner, usually hard part of a fruit that encloses the seed.
Beim Essen der Kirschen musst du auf den Kern aufpassen.
When eating the cherries, you have to watch out for the pits.

core / pit / nucleus
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pit
The inner, usually hard part of a fruit that encloses the seed.
Beim Essen der Kirschen musst du auf den Kern aufpassen.
When eating the cherries, you have to watch out for the pits.

'Kern' (core/pit/nucleus) is a very productive word. It forms many compounds such as 'Kernproblem' (core problem), 'Kernkompetenz' (core competency), or 'Kernfamilie' (nuclear family). The verb 'entkernen' means to remove the pit/core from a fruit.
Das Kernproblem unserer Firma ist der Mangel an Fachkräften.
The core problem of our company is the shortage of skilled workers.
You can recognize the figurative meaning 'wesentlicher Teil' (essential part) by the fact that 'Kern' (core) is combined with abstract nouns like 'Problem' (problem), 'Aussage' (statement), or 'Gedanke' (thought). The literal meaning refers to fruits such as cherries or peaches.
Kommen wir zum Kern der Sache.
Let's get to the core of the matter.
In its figurative meaning, 'Kern' (core) is often used in fixed expressions: 'der Kern des Problems' (the core of the problem), 'zum Kern kommen' (to get to the core), or 'im Kern' (essentially).
Deine Analyse ist im Kern richtig, aber einige Details fehlen.
Your analysis is essentially correct, but some details are missing.
The meaning 'Atomkern' (atomic nucleus) is limited to the specialist fields of physics and chemistry. Words like 'Atom' (atom), 'Energie' (energy), 'Spaltung' (fission), or 'Fusion' (fusion) in the sentence are clear signals for this meaning.
Die Kernspaltung setzt enorme Energiemengen frei.
Nuclear fission releases enormous amounts of energy.
When 'Kern' (core) describes a group of people, the adjective 'hart' (hard) is often used. 'Der harte Kern' refers to the most loyal and committed members of a group.
Der harte Kern der Fans reist zu jedem Auswärtsspiel.
The hard core of the fans travels to every away game.
middle / center
'die Mitte' and 'der Kern' can both refer to something central. 'Mitte' mainly describes location: something is roughly the same distance from the edges. 'Kern', on the other hand, often means the most important or essential part of something, that is, what it is really about. That is why the words are not always interchangeable. You can stand in the middle of a room, but the core of a problem is not simply its spatial middle. Learners may confuse the words because both are connected with 'central'. Memory tip: 'Mitte' is more about the place, while 'Kern' is more about the most important part.

Der Kern des Problems ist, dass niemand die neuen Regeln versteht.
The core of the problem is that no one understands the new rules.
Nach dem Unterricht wartet meine Nachbarin in der Mitte des Sportplatzes.
After class, my neighbor waits in the middle of the sports field.