Cold atmospheric pressure plasma jets (plasma) operated in ambient air provide a rich source of reactive oxygen and nitrogen species (RONS), which are known to influence biological processes important in disease. In the plasma treatment of diseased tissue such as subcutaneous cancer tumors, plasma RONS need to first traverse an interface between the plasma-skin surface and second be transported to millimeter depths in order to reach deep-seated diseased cells. However, the mechanisms in the plasma generation of RONS within soft tissues are not understood. In this study, we track the plasma jet delivery of RONS into a tissue model target and we delineate two processes: through target delivery of RONS generated (primarily) in the plasma jet and in situ RONS generation by UV photolysis within the target. We demonstrate that UV photolysis promotes the rapid generation of RONS in the tissue model target's surface after which the RONS are transported to millimeter depths via a slower molecular process. Our results imply that the flux of UV photons from plasma jets is important for delivering RONS through seemingly impenetrable barriers such as skin. The findings have implications not only in treatments of living tissues but also in the functionalization of soft hydrated biomaterials such as hydrogels and extracellular matrix derived tissue scaffolds.