We consider a linearly dispersing quantum impurity interacting through a contact density-density term with a one-dimensional (1D) superfluid described by the Tomonaga-Luttinger liquid theory. Using a linked-cluster expansion we characterize the impurity dynamics by calculating approximate expressions for the single-particle Green’s function and for the time evolution of the density profile. We show the existence of two different dynamical regimes: (i) a quasiparticle regime, dominated by the presence of a finite lifetime for the host impurity, and (ii) an infrared-dominated regime, where the impurity causes a Anderson’s orthogonality catastrophe of the 1D bath. We discuss the possible experimental consequences of these findings for cold-atom experiments.