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We consider a family of McKean-Vlasov equations arising as the large particlelimit of a system of interacting particles on the positive half-line withcommon noise and feedback. Such systems are motivated by structural models forsystemic risk with contagion. This contagious interaction is such that when aparticle hits zero, the impact is to move all the others toward the originthrough a kernel which smooths the impact over time. We study a rescaling ofthe impact kernel under which it converges to the Dirac delta function so thatthe interaction happens instantaneously and the limiting singularMcKean--Vlasov equation can exhibit jumps. Our approach provides a novel methodto construct solutions to such singular problems that allows for more generaldrift and diffusion coefficients and we establish weak convergence to relaxedsolutions in this setting. With more restrictions on the coefficients we canestablish an almost sure version showing convergence to strong solutions. Undersome regularity conditions on the contagion, we also show a rate of convergenceup to the time the regularity of the contagion breaks down. Lastly, we performsome numerical experiments to investigate the sharpness of our bounds for therate of convergence.

Contagious McKean--Vlasov problems with common noise: from smooth to singular feedback through hitting times