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Using the conventional T-matrix approach, we discuss gapped phases in one-, two-, and three-dimensional (3D) spin systems (both with and without a long-range magnetic order) with bond disorder and with weakly interacting bosonic elementary excitations. This work is motivated by recent experimental and theoretical activity in spin-liquid-like systems with disorder and in the disordered interacting boson problem. In particular, we apply our theory to both paramagnetic low-field and fully polarized high-field phases in dimerized spin-½ systems and in integer-spin magnets with large single-ion easy-plane anisotropy V with disorder in exchange coupling constants (and/or V). The elementary excitation spectrum and the density of states are calculated in the first order in defects concentration c ⪡ 1. In 2D and 3D systems, the scattering on defects leads to a finite damping of all propagating excitations in the band except for states lying near its edges. We demonstrate that the analytical approach is inapplicable for states near the band edges and our numerical calculations reveal their localized nature. We find that the damping of propagating excitations can be much more pronounced in considered systems than in magnetically ordered gapless magnets with impurities. In ID systems, the disorder leads to localization of all states in the band, while those lying far from the band edges (short-wavelength excitations) can took like conventional wave packets. [ABSTRACT FROM AUTHOR]