The effects of a finite substrate-helium interaction and of the crystalline structure of a substrate potential on the liquid phase of a monolayer of helium are explored for models of three systems: helium physisorbed on a basal-plane surface of graphite; He4 adsorbed on graphite preplated by a close-packed layer of neon; and He4 adsorbed on graphite preplated by a close-packed layer of argon. The ground-state energy as a function of the areal density is calculated for each of these models by introducing a class of trial wave functions which have finite extent orthogonal to the substrate, possess the same translational symmetry as the substrate, and include short-range correlations between the helium atoms. The results for He4 on bare graphite are virtually identical with previous results for the model of He4 atoms in a two-dimensional structureless box, with very small quantitative differences. The indications for He3 on bare graphite are that the liquid is not self-bound. In contrast to the bare-graphite substrate, the other two substrates exert a strong influence on the liquid, substantially increasing the equilibrium density.