We consider a simple plankton model with one phytoplankton autotroph population, one mixotrophic phytoplankton population that both photosynthesises and consumes the other phytoplankton, and one zooplankton predator population. The zooplankton predator may be either a specialist or a generalist grazer on the phytoplankton. We examine the influence on the system's equilibrium points and dynamical properties of varying degrees of mixotrophy by the phytoplankton and grazing strategy by the zooplankton, both with and without the presence of seasonal forcing. We find that the strength of the mixotrophy interaction does not substantially change the equilibrium properties of the model across all grazing strategies, but it can have a substantial effect on the dynamical properties. Further, external forcings on typical phytoplankton or zooplankton time scales can in some circumstances substantially modify these dynamics. Our analyses suggest that plankton models developed to represent bio-geochemical processes in the ocean for applications in climate change modelling should be subjected to thorough equilibrium and dynamical analyses before being used for climate prediction.