We present Atacama Large Millimeter/submillimeter Array observations of the brightest cluster galaxy Hydra-A, a nearby ($z=0.054$) giant elliptical galaxy with powerful and extended radio jets. The observations reveal CO(1-0), CO(2-1), $^{13}$CO(2-1), CN(2-1), SiO(5-4), HCO$^{+}$(1-0), HCO$^{+}$(2-1), HCN(1-0), HCN(2-1), HNC(1-0) and H$_{2}$CO(3-2) absorption lines against the galaxys bright and compact active galactic nucleus. These absorption features are due to at least 12 individual molecular clouds which lie close to the centre of the galaxy and have velocities of approximately $-50$ to $+10$ km/s relative to its recession velocity, where positive values correspond to inward motion. The absorption profiles are evidence of a clumpy interstellar medium within brightest cluster galaxies composed of clouds with similar column densities, velocity dispersions and excitation temperatures to those found at radii of several kpc in the Milky Way. We also show potential variation in a $sim 10$ km/s wide section of the absorption profile over a two year timescale, most likely caused by relativistic motions in the hot spots of the continuum source which change the background illumination of the absorbing clouds.
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