We consider Hawkes self-exciting processes with a baseline driven by an Itô semimartingale with possible jumps. Under in-fill asymptotics, we charac- terize feasible statistics induced by central limit theory for empirical average and variance of local Poisson estimates. As a byproduct, we develop a test for the absence of a Hawkes component and a test for baseline constancy. Sim- ulation studies corroborate the asymptotic theory. An empirical application on high-frequency data of the E-mini S&P500 future contracts shows that the absence of a Hawkes component and baseline constancy is always rejected.