  Recent works have shown that the contact process running on the top of highly
heterogeneous random networks is described by the heterogeneous mean-field
theory. However, some important aspects as the transition point and strong
corrections to the finite-size scaling observed in simulations are not
quantitatively reproduced in this theory. We develop a heterogeneous pair
approximation, the simplest mean-field approach that takes into account
dynamical correlations, for the contact process. The transition points obtained
in this theory are in very good agreement with simulations. The proximity with
a simple homogeneous pair-approximation is elicited showing that the transition
point in successive homogeneous cluster approximations moves away from the
simulation results. We show that the critical exponents of the heterogeneous
pair-approximation in the infinite-size limit are the same as those of the
one-vertex theory. However, excellent matches with simulations, for a wide
range of network sizes, is obtained when sub-leading finite-size corrections
given by the new theory are explicitly taken into account. The present approach
can be suited to dynamical processes on networks in general providing a
profitable strategy to analytically assess fine-tuning theoretical corrections.
