A kitten's voice may be hoarse for the following reasons:
Sore throat: Mild sore throat can cause a kitten to have a hoarse voice.
Separation from mother cat and difficulty adapting to new environment: When kittens are separated from their mother cat and come to a new environment, prolonged meowing may cause their voices to become hoarse.

Dietary issues: Insufficient water intake or improper diet in kittens can also lead to hoarseness.
Heatiness: Heatiness in kittens can also cause hoarseness.
Common cold: A cold can also cause a kitten's voice to become hoarse.