Realizing non-volatile control of superconductivity is a key step toward integrating memory and quantum functionality in future information technologies. KTaO₃-based heterostructures uniquely host interfacial two-dimensional superconductivity and quantum paraelectric lattice background. The coupling between these two degrees of freedom potentially provides a promising route to encode memory into the superconducting state. Here we reveal two intertwined phenomena in AlOₓ/KTaO₃ heterostructures: a gate-imprinted memory in which electrostatic gate cycling promotes superconductivity, and its erasure by optical illumination at cryogenic temperatures. These phenomena arise from a previously unrecognized interplay between the superconducting interface and emergent lattice excitations including polar-nanoregion reorientation and charge trapping/detrapping by oxygen vacancies. These results demonstrate configurable superconductivity at correlated oxide interfaces, opening a pathway to enrich dissipationless transport with non-volatile controls for superconducting elements.