How did the authors determine both the spatial size of gas cloud HCN-0.009-0.044 and its central mass at the same time?
The gas clouds are much larger than you think: From Fig. 1 of the paper (Takakawa 2019), the largest cloud (which they call the "Balloon") is roughly 10 arcsec across. Now they can't actually measure its exact distance, but its projected distance from the Galactic center is only 7 pc, and the authors argue that it's at least 5 kpc away due to absorption from molecular gas which is known at this distance (Sofue 2006), and since such hot and high density ($\sim 10^7\,\mathrm{cm}^{-3}$) molecular gas is abundant in the Galactic center, it is probably located there, i.e. at a distance of $R=8\,\mathrm{kpc}$.

The cloud's absolute size is thus
$$
D = 10'' \times 8\,\mathrm{kpc} = 80\,000\,\mathrm{AU} \simeq 0.4\,\mathrm{pc}.
$$

The mass is measured from dynamics, i.e. they measure the line-of-sight velocities of several clouds. These velocities depend both on the mass of the dynamical center (i.e. whatever object the clouds orbit), and on the total velocity of this object wrt. us. Since they only know the line-of-sight velocities and not the full, 3D motions, they cannot know with certainty the mass, but performing a chi-square fit of 3D models that fit the 2D observations, they determine the best-fit value of the mass to be $(3.2\pm0.6)\times10^4 M_\odot$.

Their Fig. 4 shows the model:



3D model of the motion of the clouds. The red cloud is the Balloon. Other observed clouds are shown in green, blue, and orange colors. The black dot indicates the position of the black hole. The actual observation is the 2D projection, seen in the bottom plane.

The model of the velocities also gives the actual trajectories of the clouds around the central mass. Because the radius of the Balloon's orbit is $\sim0.07\,\mathrm{pc}$, it means that this much mass must be concentrated within a region of maximum that size, and probably significantly smaller. This implies a density of $&gt;2\times10^8 M_\odot\,\mathrm{pc}^{-3}$, at least an order of magnitude larger than even the most densely packed globular clusters, and since no luminous counterpart is observed at that location, the authors consider an intermediate-mass black hole as the most promising candidate for the gravitational source.