  We explore the weak field and slow motion limits, Newtonian and
Post-Newtonian limits, of the energy-momentum powered gravity (EMPG), viz., the
energy-momentum squared gravity (EMSG) of the form
$f(T_{\mu\nu}T^{\mu\nu})=\alpha (T_{\mu\nu}T^{\mu\nu})^{\eta}$ with $\alpha$
and $\eta$ being constants. We have shown that EMPG with $\eta\geq0$ and
general relativity (GR) are not distinguishable by local tests, say, the Solar
System tests; as they lead to the same gravitational potential form, PPN
parameters, and geodesics for the test particles. However, within the EMPG
framework, $M_{\rm ast}$, the mass of an astrophysical object inferred from
astronomical observations such as planetary orbits and deflection of light,
corresponds to the effective mass $M_{\rm eff}(\alpha,\eta,M)=M+M_{\rm
empg}(\alpha,\eta,M)$, $M$ being the actual physical mass and $M_{\rm empg}$
being the modification due to EMPG. Accordingly, while in GR we simply have the
relation $M_{\rm ast}=M$, in EMPG we have $M_{\rm ast}=M+M_{\rm empg}$. Within
the framework of EMPG, if there is information about the values of
$\{\alpha,\eta\}$ pair or $M$ from other independent phenomena (from
cosmological observations, structure of the astrophysical object, etc.), then
in principle it is possible to infer not only $M_{\rm ast}$ alone from
astronomical observations, but $M$ and $M_{\rm empg}$ separately. For a proper
analysis within EMPG framework, it is necessary to describe the slow motion
condition (also related to the Newtonian limit approximation) by $|p_{\rm
eff}/\rho_{\rm eff}|\ll1$ (where $p_{\rm eff}=p+p_{\rm empg}$ and $\rho_{\rm
eff}=\rho+\rho_{\rm empg}$), whereas this condition leads to $|p/\rho|\ll1$ in
GR.
