  A second gradient generalization of Newtonian gravity is presented within the
framework of gradient field theory. Weak nonlocality is introduced via first
and second gradients of the gravitational field strength in the Lagrangian
density. Gradient generalizations of the Poisson equation of Newtonian
gravitation for the gravitational potential and of the generalized Gauss law
for the gravitational field strength are presented. Such a gradient
modification of Newtonian gravity provides a straightforward regularization of
Newtonian gravity removing the classical Newtonian singularities. Finite
gradient modifications of the gravitational potential energy and of the
gravitational force law are constructed, with a possible connection to Yukawa
interaction, and as suitable candidates for experimental tests of Newton's
inverse-square law at short distances. In addition, nonlocal gravity of
exponential type is investigated and its relation to gradient gravity theory is
given.
