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In this paper, we have constructed a WKB approximation for graphene having a Y-shaped Kekulé lattice distortion and a special folding of the <i><b>K</b></i> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mo>′</mo></msup></semantics></math></inline-formula> valleys, which leads to very specific linear energy dispersions with two non-equivalent pairs of subbands. These obtained semi-classical results, which include the action, electron momentum and wave functions, are utilized to analyze the dynamics of electron tunneling through non-square potential barriers. In particular, we explore resonant scattering of an electron by a potential barrier built on Kekulé-distorted graphene. Mathematically, a group of consecutive equations for a semi-classical action have been solved by following a perturbation approach under the condition of small strain-induced coupling parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>Δ</mo><mn>0</mn></msub><mo>≪</mo><mn>1</mn></mrow></semantics></math></inline-formula> (a good fit to its actual value <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>Δ</mo><mn>0</mn></msub></semantics></math></inline-formula>∽ 0.1). Specifically, we consider a generalized model for Kek-Y graphene with two arbitrary Fermi velocities. The dependence of the electron transmission amplitude on the potential profile <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> and band parameters of Kekulé-patterned graphene has been explored and analyzed in detail.