  We derive the full set of universal relations for spin-polarized Fermi gases
with $p$-wave interaction in two dimensions, simply using the short-range
asymptotic behavior of fermion-pair wave functions. For $p$-wave interactions,
an additional contact related to the effective range needs to be introduced,
besides the one related to the scattering volume. Since the subleading tail
($k^{-4}$) of the large-momentum distribution cannot fully be captured by the
contacts defined by the adiabatic relations, an extra term resulted from the
center-of-mass motions of the pairs gives rise to an additional divergence in
the kinetic energy of the system, besides those related to the contacts
defined. We show in Tan's energy theorem that if only two-body correlations are
taken into account, all these divergences are reasonably removed, leading to a
finite internal energy of the system. In addition, we find that all the other
universal relations, such as the high-frequency behavior of the radio-frequency
response, short-range behavior of the pair correlation function, generalized
virial theorem, and pressure relation, remain unaffected by the center-of-mass
motions of the pairs, and are fully governed by the contacts defined by the
adiabatic relations. Our results confirm the feasibility of generalizing the
contact theory for higher-partial-wave scatterings, and could readily be
confirmed in current experiments with ultracold $^{40}$K and $^{6}$Li atoms.
