Suppose that the integers are assigned the random variables $\{\omega_x,\mu_x\}$ (taking values in the unit interval times the space of probability measures on $\reals_+$), which serve as an environment. This environment defines a random walk $\{X_t\}$ (called a RWREH) which, when at x, waits a random time distributed according to $\mu_x$ and then, after one unit of time, moves one step to the right with probability $\omega_x$, and one step to the left with probability $1-\omega_x$. We prove large deviation principles for $X_t/t$, both quenched (i.e., conditional upon the environment), with deterministic rate function, and annealed (i.e., averaged over the environment). As an application, we show that for random walks on Galton--Watson trees, quenched and annealed rate functions along a ray differ.