  Cool clouds are expected to be destroyed and incorporated into hot
supernova-driven galactic winds. The mass-loading of a wind by the cool medium
modifies the bulk velocity, temperature, density, entropy, and abundance
profiles of the hot phase relative to an un-mass-loaded outflow. We provide
general equations and limits for this physics that can be used to infer the
rate of cool gas entrainment from X-ray observations, accounting for
non-spherical expansion. In general, mass-loading flattens the density and
temperature profiles, decreases the velocity and increases the entropy if the
Mach number is above a critical value. We first apply this model to a recent
high-resolution galactic outflow simulation where the mass-loading can be
directly inferred. We show that the temperature, entropy, and composition
profiles are well-matched, providing evidence that this physics sets the bulk
hot gas profiles. We then model the diffuse X-ray emission from the local
starburst M82. The non-spherical (more cylindrical) outflow geometry is
directly taken from the observed X-ray surface brightness profile. These models
imply a total mass-loading rate that is about equal to that injected in the
starburst, $\simeq 10$ M$_\odot$ yr$^{-1}$, and they predict an asymptotic hot
wind velocity of $\sim 1000\,{\rm km \ s^{-1}}$ that is $\sim1.5-2$ times
smaller than previous predictions. We also show how the observed entropy
profile can be used to constrain the outflow velocity, making predictions for
future missions like XRISM. We argue that the observed X-ray limb-brightening
may be explained by mass-loading at the outflow's edges.
