Why wouldn't "Puni manakō nā hoa hānau" work?
That's seems like, 'the cousins love mango'.
In this context, does it mean the fruit mango or just the flavor in general?
Good question! It could be either. The meaning is ambiguous in that respect.
Why is the "i" not in front of the cousins?
"ka manakō" is the object of the sentence, so the "i" comes in front of that part. It might be helpful to think about it this way: the cousins are the ones doing the liking "to" the mango.
The correct word, hoa, was not available in the selection possibilties.