Intuitively, specifying a small number of integer points in a small
region of some polytope has almost no connection with the rest of the
polytope, so the answer should be negative. To give a concrete example,
let $P$ be an $n$-element poset and $\mathcal{O}(P)$ its order
polytope (see http://math.mit.edu/~rstan/pubs/pubfiles/66.pdf). If you
are given the facets of $\mathcal{O}(P)$ then computing the number of
integer points is equivalent to computing the number $j(P)$ of order ideals
of $P$ (which is at most $2^n$), and is known to be
$\#P$-complete. Suppose $P$ is the disjoint union of an $n/2$-element
antichain (say) and an $n/2$-element poset $Q$. Then we know the
$(n/2)!$ order ideals of the antichain, which is far more than $\log_2
j(P)\leq n$. Moreover, $j(Q)=(n/2)!j(Q)$, and computing $j(Q)$ is
$\#P$-complete. I doubt whether specifying any $n$ (or some much
larger number) of order ideals of $P$ will be of much help in counting
the total number of them.