As far as I understand HMC, it does not address the problem of avoiding repeated computations. Instead, it provides a way to utilize knowledge of the derivative of the target (likelihood/score/posterior/measure/…) function to propose efficient sampling update steps. This has the potential to avoid many attempts to sample “in uninteresting directions” and in this way can speedup sampling. As far as I am aware, in the context of statistical physics and Bayesian parameter inference where HMC is used, the idea of reusing partial evaluations of the target function is not widespread. The target function is regarded as a function, not so much as a probabilistic program.
If I’m not up to date, I’d love to learn otherwise!