Spin -split superconductors offer new functionality compared to conventional superconductors such as diode effects and efficient thermoelectricity. The superconducting state can nevertheless only withstand a small amount of spin splitting. Here, we self -consistently determine the spin transport properties and the phase diagram of a spin -split superconductor in the presence of an injected spin accumulation. Energy and spin relaxation are accounted for in the relaxation time approximation via a single effective inelastic scattering parameter. We find that the spin -splitting field in the superconductor enables a spin diode effect. Moreover, we consider the superconducting phase diagram of a system in contact with a spin accumulation and in the presence of spin relaxation, and find that the inclusion of energy and spin relaxation alters the phase diagram qualitatively. In particular, these mechanisms turn out to induce a superconducting state in large parts of the phase diagram where a normal state would otherwise be the ground state. We identify an Fulde-Ferrel-Larkin-Ovchinnikkov-like state even in the presence of impurity scattering, which can be controllably turned on and off via the electrically induced spin accumulation. We explain the underlying physics from how the superconducting order parameter depends on the nonequilibrium modes in the system as well as the behavior of these modes in the presence of energy and spin relaxation when a spin -splitting field is present.