What is meant by Category theory doesn't yet know how to deal with higher-order functions?
The issue with higher-order functions is simple enough to state.  


A type-constructor like $T(X) = [X \to X]$ is not a functor.  It should have been.  
A polymorphic function like ${\it twice}_X : T(X) \to T(X) = \lambda f.\, f \circ f$ is not a natural transformation.  It should have been.


If you read Eilenberg and MacLane's original category theory paper, (PDF) the intuitions they present cover those cases.  But their theory doesn't.  Theirs was a great paper for 1945!  But, today, we need more.

The reaction of category theorists to these issues is a bit perplexing.  They act as if higher-order operations form a Computer Science idea; they are of no consequence to mathematics.  If that is so, then a foundation of mathematics would not be good enough for a foundation of computer science.

But I don't seriously believe that.  I believe that higher-order functions would be quite important for mathematics as well.  But they have not been seriously explored.  I am hopeful that, some day, they will be explored and the limitations of category theory will be realized.