Learn Turkish: Öğretmen soruyu tekrar etti. - The teacher repeated the question.

Auxiliary verb compounds with etmek or olmak are written as a single word only if the base noun undergoes a sound change. If the noun doubles its final consonant (like his becoming hissetti) or drops a vowel (like şükür becoming şükretti), they combine. Since tekrar doesn't change its spelling when paired with an auxiliary, the two words stay separate.

Why does soru take the -(y)u ending here?

The verb compound tekrar etmek is transitive and here applies to a specific, known object (the question). Therefore, soru (question) needs the definite accusative case. Since it ends in a vowel, it takes the buffer consonant -y- followed by -u to satisfy four-way vowel harmony.

The root is et-, so why doesn't the t soften to d here like it does in ediyor?

The t in etmek only softens to d (consonant mutation) when followed by a suffix starting with a vowel. Because the past tense suffix starts with a consonant, the root remains et-. In fact, the voiceless t of the root forces the suffix to harden, changing the past tense marker from -di to -ti.