This　paper　investigates　a　Leslie-Gower　predator-prey　model　with　simplified　Holling　type　IV　functional　response　and　constant-yield　prey　harvesting.　We　analyze　conditions　for　the　existence　of　positive　equilibria　and　prove　that　the　system　has　at　most　four　positive　equilibria.　The　results　show　that　the　double　positive　equilibrium　is　a　cusp　of　codimension　at　most　4,　the　triple　positive　equilibrium　is　a　degenerate　saddle　or　nilpotent　focus　of　codimension-3,　and　the　quadruple　positive　equilibrium　is　a　nilpotent　cusp　of　codimension-5.　In　addition,　as　the　parameters　vary,　the　system　can　undergo　a　cusp-type　(or　focus-type)　degenerate　Bogdanov-Takens　bifurcation　of　codimension-4　(or　codimension-3).　Furthermore,　the　positive　equilibrium　is　a　weak　focus　of　order　at　most　4,　and　the　model　can　undergo　a　degenerate　Hopf　bifurcation　of　codimension-4.　Finally,　our　main　results　are　verified　by　some　numerical　simulations,　which　also　reveal　that　there　exist　three　limit　cycles　containing　one　positive　equilibrium,　or　one　(or　two)　limit　cycles　containing　three　positive　equilibria,　or　a　limit　cycle　as　well　as　a　homoclinic　loop.　　©　2025　World　Scientific　Publishing　Company.