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鳴く seems to be the bark, cry, etc. of animals, and 泣く seems to be crying, weeping, etc. of humans (at any rate, associated with pain, sadness, etc.). However, when a dog in pain whines or howls, which of the two will be used?
Technically speaking, that would still be 「鳴く」.
But as an animal lover or pet owner in that particular situation, you might personify the dog out of empathy and end up choosing 「泣く」. That would not be considered an "incorrect" usage.
Required, but never shown