Here we show that asymmetric fully localized flexural-gravity lumps can propagate on the surface of an inviscid and irrotational fluid covered by a variable-thickness elastic material, provided that the thickness varies only in one direction and has a local minimum. We derive and present equations governing the evolution of the envelope of flexural-gravity wave packets allowing the flexing material to have small variations in the transverse (to propagation) direction. We show that the governing equation belongs to the general family of Davey-Stewartson equations, but with an extra term in the surface evolution equation that accounts for the variable thickness of the elastic cover. We then use an iterative Newton-Raphson scheme, with a numerical continuation procedure via Lagrange interpolation, in a search to find fully localized solutions of this system of equations. We show that if the elastic sheet thickness has (at least) a local minimum, flexural-gravity lumps can propagate near the minimum thickness, and in general have an asymmetric bell-shape in the transverse to the propagation direction. In applied physics, flexural-gravity waves describe for instance propagation of waves over the ice-covered bodies of water. Ice is seldom uniform, nor is the seafloor, and in fact near the boundaries (ice-edges, shorelines) they typically vary only in one direction (toward to edge), and are uniform in the transverse direction. This research suggests that fully localized waves are not restricted to constant ice-thickness/water-depth areas and can exist under much broader conditions. Presented results may have implications in experimental generation and observation of flexural-gravity (as well as capillary-gravity) lumps.