  Under various conditions on the data we analyse how appearence of lower order
terms affects the gradient estimates on solutions to a general nonlinear
elliptic equation of the form \[-{\rm div}\, a(x,Du)+b(x,u)=\mu\] with data
$\mu$ not belonging to the dual of the natural energy space but to
Lorentz/Morrey-type spaces. The growth of the leading part of the operator is
governed by a function of Orlicz-type, whereas the lower-order term satisfies
the sign condition and is minorized with some convex function, whose speed of
growth modulates the regularization of the solutions.
