  First, we show that if the pressure $p$ (associated to a weak Leray-Hopf
solution $v$ of the Navier-Stokes equations) satisfies
$\|p\|_{L^{\infty}_{t}(0,T^*; L^{\frac{3}{2},\infty}(\mathbb{R}^3))}\leq M^2$,
then $v$ possesses higher integrability up to the first potential blow-up time
$T^*$. Our method is concise and is based upon energy estimates applied to
powers of $|v|$ and the utilization of a `small exponent'.
  As a consequence, we show that if a weak Leray-Hopf solution $v$ first blows
up at $T^*$ and satisfies the Type I condition $\|v\|_{L^{\infty}_{t}(0,T^*;
L^{3,\infty}(\mathbb{R}^3))}\leq M$, then $$\nabla v\in
L^{2+O(\frac{1}{M})}(\mathbb{R}^3\times (\tfrac{1}{2}T^*,T^*)).$$ This is the
first result of its kind, improving the integrability exponent of $\nabla v$
under the Type I assumption in the three-dimensional setting.
  Finally, we show that if $v:\mathbb{R}^3\times [-1,0]\rightarrow
\mathbb{R}^3$ is a weak Leray-Hopf solution to the Navier-Stokes equations with
$s_{n}\uparrow 0$ such that
$$\sup_{n}\|v(\cdot,s_{n})\|_{L^{3,\infty}(\mathbb{R}^3)}\leq M $$ then $v$
possesses at most $O(M^{20})$ singular points at $t=0$. Our method is direct
and concise. It is based upon known $\varepsilon$-regularity, global bounds on
a Navier-Stokes solution with initial data in $L^{3,\infty}(\mathbb{R}^3)$ and
rescaling arguments. We do not require arguments based on backward uniqueness
nor unique continuation results for parabolic operators.
