  We give almost-linear-time algorithms for constructing sparsifiers with $n\
poly(\log n)$ edges that approximately preserve weighted $(\ell^{2}_2 +
\ell^{p}_p)$ flow or voltage objectives on graphs. For flow objectives, this is
the first sparsifier construction for such mixed objectives beyond unit
$\ell_p$ weights, and is based on expander decompositions. For voltage
objectives, we give the first sparsifier construction for these objectives,
which we build using graph spanners and leverage score sampling. Together with
the iterative refinement framework of [Adil et al, SODA 2019], and a new
multiplicative-weights based constant-approximation algorithm for
mixed-objective flows or voltages, we show how to find $(1+2^{-\text{poly}(\log
n)})$ approximations for weighted $\ell_p$-norm minimizing flows or voltages in
$p(m^{1+o(1)} + n^{4/3 + o(1)})$ time for $p=\omega(1),$ which is almost-linear
for graphs that are slightly dense ($m \ge n^{4/3 + o(1)}$).
