Obtaining an acyclic graph by removing edges using an algorithm that decides ACYCLIC
Disclaimer This solution assumes that the language $\text{Acyclic}$ is the language that contains exactly all acyclic directed graphs.

It is impossible to achieve this in polynomial time unless $\operatorname{P}=\operatorname{NP}$. The reason is that the problem you have to solve is NP-hard. It is called the directed feedback arc set problem. It is one of Karp's 21 hard problems. On the other hand, finding whether a graph is acyclic can be done with any graph traversal method in polynomial (actually linear) time. Hence, $\operatorname{P}^{\text{Acyclic}} = \operatorname P$. Hence, if you solved the task you are given in polynomial time you would have proven $\operatorname{P} = \operatorname{NP}$.