Horowitz-Langevin MCMC is an extension of MALA (or HMC with only one leapfrog step) that changes the rules of updating the momentum. By preserving the momentum to some degree, Horowitz-Langevin MCMC tries to overcome the random walk behavior of the Langevin MCMC in regions where the posterior falls off quickly in most directions. By preserving the momentum of accepted proposals to some degree, the Horowitz-Langevin MCMC explores the high probability regions more efficiently.

See
[1] Horowitz, A. M. (1991) “A generalized guided Monte Carlo algorithm”, Physics Letters B, vol. 268, pp. 247-252.
[2] https://arxiv.org/abs/2001.11950v1