We consider a covariant theory of gravity defined by the action S = ∫ d^4x √−g [(16πG)^{-1}(R + αR^2) + L_matter], which reduces to general relativity when α → 0. This theory, belonging to the class of f(R) gravities, is ghost-free and contains an additional massive scalar degree of freedom (the scalaron), and is treated as an effective field theory valid below the Planck scale. We derive the fourth-order field equations step by step, including all boundary terms. We then apply them to a static, spherically symmetric metric describing a traversable wormhole. The metric includes a Gaussian deformation X(r) = A exp[−(r − r0)^2/(2σ^2)] localised around the throat r = r0. The flare-out condition is satisfied for A > −1. We compute all Christoffel symbols exactly, derive the Ricci tensor and scalar, and solve the modified Einstein equations for an anisotropic fluid. The energy conditions are analysed: the null energy condition is violated only in an arbitrarily small region near the throat if α > 0 and the Gaussian parameters are chosen appropriately. We study null geodesics using the corrected radial equation, compute the deflection angle and the Shapiro time delay for light passing near the wormhole, and show that the Gaussian deformation introduces a characteristic shift compared to the Morris—Thorne case. Stability under radial perturbations is analysed in the scalar—tensor representation, showing that the wormhole can be stable for a range of parameters. The asymptotic limit is studied via a post-Newtonian expansion: the PPN parameters remain γ = β = 1, and the scalaron mass m^2 = 1/(6α) is constrained by Solar System tests to be m ≥ 10^{-3} eV. We discuss observational signatures such as lensing, Shapiro delay, and gravitational-wave echoes that make the model falsifiable. All calculations are presented in detail, with intermediate steps collected in appendices.