Learn Polish: Przez moją nieostrożność przegapiłem ważną wiadomość. - Because of my carelessness I missed an important message.

In Polish, przez + Accusative is the most natural way to express fault, blame, or a negative cause. While z powodu (+ Genitive) also means "because of" or "due to" and would be grammatically correct here, przez specifically emphasizes that someone or something is to blame. (Remember never to use dzięki here, as that means "thanks to" and is strictly for positive outcomes).

Przez requires the Accusative case. Why does the noun nieostrożność look like it hasn't changed from the Nominative?

It actually is in the Accusative! Nieostrożność (carelessness) is a feminine noun, as are almost all abstract Polish nouns ending in -ość. A very helpful rule to remember is that feminine nouns ending in a consonant have the exact same form in the Accusative case as they do in the Nominative. You can tell the case has changed because the pronoun attached to it correctly takes the feminine Accusative ending, becoming moją.

Could I use the verb tęskniłem instead of przegapiłem to say "I missed"?

No, because they refer to completely different types of "missing". Tęsknić means to miss someone or something emotionally (like missing your family, your hometown, or the good old days). Przegapić is used when you fail to notice, catch, or attend something—like missing a bus, overlooking a deadline, or failing to see a message.

Why does the adjective end in if its noun doesn't change its ending?