A number of ways to describe a hoarse or rough voice come to mind.
We say someone has a husky voice. Yes, same spelling as the dog. No, it doesn't mean they sound like a dog or stare at you with a doge smile. It means their voice is hoarse and dry. It could be their natural voice or a result of illness or emotion.
Examples: See YouTube examples.
This girl has a husky voice coming back from a protest, saying of herself "(I am) a sick person with the sexiest voice around."
I'm sorry my voice is husky from a cold.
Also you could use throaty, croaky. I would say throaty implies a deepness in sound, coming from the throat. Croaky similarly suggests the voice is low and deep.
I'm sorry my voice is kind of throaty/croaky today from a cold.
More commonly, raspy also works.
Hey did you hear the teacher today in class? Her voice was so raspy.
A comment said these word are "all literary". I disagree. There is nothing literary about "raspy" or "husky". Added some examples.
The word husky has another polysemous adjective meaning burly and beefy, used to describe (usually) men's physique. It has influenced the other meaning of a kind of sound marked by hoarseness. It is sometimes associated with a male voice, but so are hoarse, deep and low voices.