We generalize an inequality of E. Heintze and H. Karcher for the volume of tubes around minimal submanifolds to an inequality based on integral bounds for k-Ricci curvature. Even in the case of a pointwise bound this generalizes the classical inequality by replacing a sectional curvature bound with a k-Ricci bound. This work is motivated by the estimates of Petersen-Shteingold-Wei for the volume of tubes around a geodesic and generalizes their estimate. Using similar ideas we also prove a Hessian comparison theorem for k-Ricci curvature which generalizes the usual Hessian and Laplacian comparison for distance functions from a point and give several applications.