  In a recent experiment [P. Peng, $et$ $al.$, Phys. Rev. A \textbf{97}, 012702
(2018)], it has been shown that the $p$-wave Feshbach resonance can be shifted
toward the $s$-wave Feshbach resonance by a laser field. Based on this
experiment, we study the universal relations and the normal-state properties in
an ultracold Fermi gas with coexisting $s$- and $p$-wave interactions under
optical control of a $p$-wave magnetic Feshbach resonance. Within the
operator-product expansion, we derive the high-momentum tail of various
observable quantities in terms of contacts. We find that the high-momentum tail
becomes anisotropic. Adopting the quantum virial expansion, we calculate the
normal-state contacts with and without a laser field for $^{40}$K atoms using
typical experimental parameters. We show that the contacts are dependent on the
laser dressing. We also reveal the interplay of laser dressing and different
partial-wave interactions on various contacts. In particular, we demonstrate
that the impact of the laser dressing in the $p$-wave channel can be probed by
measuring the $s$-wave contacts, which is a direct manifestation of few-body
effects on the many-body level. Our results can be readily checked
experimentally.
