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This paper extends to transonic mixing layers an analysis of Lighthill ("Reflection at a Laminar Boundary Layer of a Weak Steady Disturbance to a Supersonic Stream, Neglecting Viscosity and Heat Conduction," Quarterly Journal of Mechanics and Applied Mathematics, Vol. 54, No. 3, 1950, pp. 303-325.) on the interaction between weak shocks and laminar boundary layers. As in that work, the analysis is carried out under linear-inviscid assumptions for the perturbation field, with streamwise changes of the base flow neglected, as is appropriate given the slenderness of the mixing-layer flow. The steady-disturbance profile is determined by taking a Fourier transform along the longitudinal coordinate. Closed-form analytical functions for the pressure field are derived in the small- and large-wave-number limits, and vorticity disturbances are obtained as functions of the pressure perturbations. The analysis is particularized to ethylene&-air and hydrogen&-air mixing layers, for which the dynamics are of current interest for hypersonic propulsion. The results provide, in particular, the effective distance of upstream influence of the pressure perturbation in the subsonic stream. The resulting value, which scales with the thickness of the subsonic layer, is much smaller than the upstream influence distances encountered in boundary layers. This study may serve as a basis to understand shock-induced autoignition and flameholding phenomena in simplified versions of non-premixed supersonic-combustion problems.