Killing weights from the perspective of t-structures

This paper is devoted to morphisms killing weights in a range (as defined by the first author) and to objects without these weights (as essentially defined by J. Wildeshaus) in a triangulated category endowed with a weight structure w. We describe several new criteria for morphisms and objects to satisfy these conditions. In some of them we use virtual t-truncations and a t-structure adjacent to w. In the case where the latter exists we prove that a morphism kills weights $m,...,n$ if and only if it factors through an object without these weights; we also construct new families of torsion theories and projective and injective classes. As a consequence, we obtain some "weakly functorial decompositions" of spectra (in the stable homotopy category SH) and a new description of those morphisms that act trivially on degree zero singular cohomology with coefficients in every abelian group.