A　Leslie-Gower　predator-prey　model　with　nonlinear　harvesting　and　a　generalist　predator　is　considered　in　this　paper.　It　is　shown　that　the　degenerate　positive　equilibrium　of　the　system　is　a　cusp　of　codimension　up　to　4,　and　the　system　admits　the　cusp-type　degenerate　Bogdanov-Takens　bifurcation　of　codimension　4.　Moreover,　the　system　has　a　weak　focus　of　at　least　order　3　and　can　undergo　degenerate　Hopf　bifurcation　of　codimension　3.　We　verify,　through　numerical　simulations,　that　the　system　admits　three　different　stable　states,　such　as　a　stable　fixed　point　and　three　limit　cycles　(the　middle　one　is　unstable),　or　two　stable　fixed　points　and　two　limit　cycles.　Our　results　reveal　that　nonlinear　harvesting　and　a　generalist　predator　can　lead　to　richer　dynamics　and　bifurcations　(such　as　three　limit　cycles　or　tristability);　specifically,　harvesting　can　cause　the　extinction　of　prey,　but　a　generalist　predator　provides　some　protection　for　the　predator　in　the　absence　of　prey.