In this paper, we point out that when p is a prime, the additive structure of the Hochschild cohomology ring of a p-block of a finite group G satisfies the now-familiar type of alternating sum formula using the p-subgroup complex of G. (Throughout, we work with the Tate version of Hochschild cohomology, which coincides with ordinary Hochschild cohomology in positive dimensions.) Such an expression (in our situation) allows the Hochschild cohomology of the block to be computed from p-local information. The first such formula (for the cohomology of a G-module) was obtained by P. J. Webb (see, for example, [6]). We also give a formulation of our result in terms of subpairs. For illustrative purposes, we give an application in the situation when the block is controlled by the normalizer of a maximal subpair, and the outer automorphims of the defect group induced by the inertial quotient all act without nontrivial fixed-points. This application generalizes an earlier result of Kessar and Linckelmann [2], which dealt with such a situation in the case that the defect group is Abelian.