Is this phrase correct? "Una ave bonita"

Hola Trefor

With the article un/una both are considered correct as mentioned in the lesson. It is with el/la that only the masculine form is admitted: el ave, un ave, una ave. 

Saludos

Hola Trefor

The reason to use el and not la in front of these words is to make it easier for their pronunciation, so there aren't two "a" in a row "la agua" - it's easier to say "el agua". 

As for the reason to admit both un and una, there is no known explanation and the RAE doesn't expand on it. Although it's a lot more common to say "un ave", the Academy doesn't penalise "una ave" (but it is less common). 

Here's an explanation of what they say.

Saludos