A theory on incoherently coupled low-amplitude soliton pairs for screening-photovoltaic solitons in biased photorefractive-photovoltaic crystals is investigated that gives rise to the analytical solutions of bright-bright and dark-dark soliton pairs, which are due to both the spatially nonuniform screening of the external electric field and the photovoltaic effect. These soliton pairs, differ from screening soliton pairs originating from the spatially nonuniform screening of the external electric field and photovoltaic soliton pairs originating from the photovoltaic effect, are of interesting properties. When the photovoltaic effect is neglected, their nonlinear wave equations change into that for screening solitons, and these soliton pairs change into screening soliton pairs. When the external bias field is absent, their nonlinear wave equation change into that for photovoltaic solitons in closed and open circuits, and these soliton pairs change into photovoltaic soliton pairs in closed and open circuits. If the carrier beams share the same polarization, wavelength, and are mutually incoherent, these soliton pairs can be obtained. Relevant examples are provided where the photorefractive-photovoltaic crystal is lithium niobate (LiNbO3) crystal.