  We analyze multi-dimensional mean-field stochastic differential equations
where the drift depends on the law in form of a Lebesgue integral with respect
to the pushforward measure of the solution. We show existence and uniqueness of
Malliavin differentiable strong solutions for irregular drift coefficients,
which in particular include the case where the drift depends on the cumulative
distribution function of the solution. Moreover, we examine the solution as a
function in its initial condition and introduce sufficient conditions on the
drift to guarantee differentiability. Under these assumptions we then show that
the Bismut-Elworthy-Li formula proposed in Bauer et al. (2018) holds in a
strong sense, i.e. we give a probabilistic representation of the strong
derivative with respect to the initial condition of expectation functionals of
strong solutions to our type of mean-field equations in one-dimension.
