  We present a detailed synchrotron x-ray scattering study of the
charge-density-wave (CDW) order in simple tetragonal HgBa$_2$CuO$_{4+\delta}$
(Hg1201). Resonant soft x-ray scattering measurements reveal that short-range
order appears at a temperature that is distinctly lower than the pseudogap
temperature and in excellent agreement with a prior transient reflectivity
result. Despite considerable structural differences between Hg1201 and
YBa$_2$Cu$_3$O$_{6+\delta}$, the CDW correlations exhibit similar doping
dependencies, and we demonstrate a universal relationship between the CDW wave
vector and the size of the reconstructed Fermi pocket observed in quantum
oscillation experiments. The CDW correlations in Hg1201 vanish already below
optimal doping, once the correlation length is comparable to the CDW modulation
period, and they appear to be limited by the disorder potential from unit cells
hosting two interstitial oxygen atoms. A complementary hard x-ray diffraction
measurement, performed on an underdoped Hg1201 sample in magnetic fields along
the crystallographic $c$ axis of up to 16 T, provides information about the
form factor of the CDW order. As expected from the single-CuO$_2$-layer
structure of Hg1201, the CDW correlations vanish at half-integer values of $L$
and appear to be peaked at integer $L$. We conclude that the atomic
displacements associated with the short-range CDW order are mainly planar,
within the CuO$_2$ layers.
