The Hawks process is a point process with a self-exciting property. It has been used to model earthquakes, social media events, infections, etc., and is getting a lot of attention. However, as a real problem, there are often situations where we can not obtain data with sufficient observation time. In such cases, it is not appropriate to approximate the error distribution of an estimator by the normal distribution. To overcome this problem, we provide a rigorous mathematical foundation of the theory for the higher-order asymptotic behavior of the one-dimensional Hawkes process with an exponential kernel. As an important application, we give the second-order asymptotic distribution for the maximum likelihood estimator of the exponential Hawkes process. Furthermore, we also present the simulation results.