"The girl wants to have a cat."
Translation:Pigen vil gerne have en kat.
No. "Pigen vil have en kat" could be a demand or a wish. "Pigen vil gerne have en kat" could only be a wish, and it would be a more polite way of expressing it than the version without "gerne".
Your translation is grammatically incorrect because it needs to be "vil gerne +infinitive".