  We prove ``effective'' linear response for certain classes of non-uniformly
expanding random dynamical systems which are not necessarily composed in an
i.i.d manner. In applications, the results are obtained for base maps with a
sufficient amount of mixing. The fact that the rates are effective is then
applied to obtain the differentiability of the variance in the CLT as a
function of the parameter, as well as the annealed linear response. These two
applications are beyond the reach of the linear response obtained in the
general case, when all the random variables appearing in the bounds are only
tempered. We also provide several wide examples of one-dimensional maps
satisfying our conditions, as well as some higher-dimensional examples.
