A　Leslie-Gower　predator-prey　model　with　Smith　growth　and　constant-yield　harvesting　is　proposed　in　this　paper.　We　show　that　the　system　admits　at　most　two　boundary　equilibria,　both　of　which　are　unstable.　The　degenerate　positive　equilibrium　of　the　system　is　a　cusp　of　codimension　2,　and　the　system　undergoes　cusp-type　Bogdanov-Takens　bifurcation　of　codimension　2.　Moreover,　we　prove　that　the　system　has　a　weak　focus　of　order　at　most　3,　and　the　system　can　undergo　a　degenerate　Hopf　bifurcation　of　codimension　3.　Our　results　reveal　that　the　constant-yield　harvesting　can　lead　to　richer　dynamic　behaviors.　©　2024　the　Author(s),　licensee　AIMS　Press.　This　is　an　open　access　article　distributed　under　the　terms　of　the　Creative　Commons　Attribution　License　(https://creativecommons.org/licenses/by/4.0)