We derive an on-shell representation for electroweak 2 + J → 2 transition amplitudes involving two hadrons in both the initial and final state for systems where there are left-hand singularities generated by light-particle exchange, e.g. one-pion exchange. The derivation treats the insertion of the electroweak current perturbatively, keeping only the leading-order contribution in the external field, while the hadronic interactions are treated to all orders, including one-particle exchange effects. We find that, in such processes, the amplitude can have logarithmic singularities, as well as one-particle pole singularities, due to these exchanges. We isolate those contributions, as well as previously identified triangle singularities. The result is expressed in terms of the purely hadronic amplitude, exchange-current kernels, triangle functions, and a class of short-distance transition functions that are real and smooth below unaccounted-for thresholds. While we consider only transitions with spinless particles, this work is an important step toward constraining form factors of systems involving nucleons or vector mesons in the heavy quark sector, where the one-pion-exchange singularity is quite close to threshold.