  In this paper, we investigate systematically the supercritical conditions on
the pressure $\pi$ associated to a Navier-Stokes solution $v$ (in
three-dimensions), which ensure a reduction in the Hausdorff dimension of the
singular set at a first potential blow-up time. As a consequence, we show that
if the pressure $\pi$ satisfies the endpoint scale invariant conditions
$$\pi\in
L^{r,\infty}_{t}L^{s,\infty}_{x}\quad\textrm{with}\,\,\tfrac{2}{r}+\tfrac{3}{s}=2\,\,\textrm{and}\,\,r\in
(1,\infty),$$ then the Hausdorff dimension of the singular set at a first
potential blow-up time is arbitrarily small. This hinges on two ingredients:
(i) the proof of a higher integrability result for the Navier-Stokes equations
with certain supercritical assumptions on $\pi$ and (ii) the establishment of a
convenient $\varepsilon$- regularity criterion involving space-time integrals
of $$|\nabla v|^2|v|^{q-2}\,\,\,\textrm{with}\,\,q\in (2,3). $$ The second
ingredient requires a modification of ideas in Ladyzhenskaya and Seregin's
paper, which build upon ideas in Lin, as well as Caffarelli, Kohn and
Nirenberg.
