Nature presents intriguing biological swimmers with innate energy harvesting abilities from their local environments. Use of natural swimmers as cargo delivery agents presents an alternative strategy to transport therapeutics inside the body to locations otherwise difficult-to-access by traditional delivery strategies. Even though bacteria are heavily utilized as actuators in biohybrid microswimmer designs for active cargo delivery applications, their possible acute pathogenicity necessitates search for an agile biological swimmer with better biocompatibility, such as microalgae. In this project, we report a biocompatible biohybrid microswimmer powered by a unicellular freshwater green microalga, Chlamydomonas reinhardtii. Polyelectrolyte-functionalized magnetic spherical cargoes (1 $\mu$m in diameter) were attached to surface of the microalgae via non-covalent interactions without the requirement for any chemical reaction. Three-dimensional swimming motility of the constructed biohybrid algal microswimmers was characterized in the presence and absence of a uniform magnetic field in the x-direction. In addition, motility of both microalgae and biohybrid algal microswimmers was investigated in various physiologically relevant conditions, including cell culture medium, human tubal fluid, plasma and blood. Furthermore, it was demonstrated that the algal microswimmers are cytocompatible when co-cultured with certain healthy and cancerous cells. Finally, fluorescent isothiocyanate-dextran (a water-soluble polysaccharide) molecules were effectively delivered to mammalian cells using the biohybrid algal microswimmers as a proof-of-concept active cargo delivery demonstration. The microswimmer design described here presents a new class of biohybrid microswimmers with greater biocompatibility and motility for targeted delivery applications in medicine.