In　this　paper,　we　revisit　a　Leslie　type　predator–prey　system　with　strong　Allee　effect　on　prey　and　simplified　Holling　type　IV　functional　response　proposed.　Some　more　complex　dynamical　properties　have　been　discovered.　First,　it　is　proved　that　the　system　exhibits　a　degenerate　cusp　of　codimension　4　and　a　degenerate　focus,　saddle　or　elliptic　of　codimension　3　for　different　parameter　values　analytically.　Further,　various　possible　high　codimensional　bifurcation　analyses　are　performed.　It　is　shown　that　the　system　undergoes　degenerate　focus,　saddle　or　elliptic　type　Bogdanov–Takens　bifurcation　of　codimension　3　and　degenerate　cusp　type　Bogdanov–Takens　bifurcation　of　codimension　4　as　the　parameters　vary.　Moreover,　the　existence　and　spatial　location　of　the　limit　cycles　are　explored　by　calculating　Hopf　bifurcation　and　homoclinic　bifurcation　surfaces.　Numerical　simulations　are　carried　out,　and　it　is　observed　that　three　limit　cycles　coexist.　©　2023,　The　Author(s),　under　exclusive　licence　to　Springer　Nature　Switzerland　AG.