It's a good question, and you are right that it was not fully answered in the other related question.
The answer was that it is due to the Einstein Field Equations (EFE) for General Relativity. That is in @QMechanics reference to How energy curves spacetime?
And that is true as far as it goes. The EFE say that spacetime is a dynamic entity, and it's curvature is determined by the matter-energy in that spacetime. The matter-energy is defined by a stress energy tensor which includes anything with momentum, anything with stress, and anything with energy. Since by E = $mc^2$ mass has an equivalence to energy, it contributes to the energy term. So the EFEs are a covariant set of equations that relate, in any coordinate systems, the Einstein tensor to the stress energy tensor as
$G_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi T_{\mu\nu}$
where the indices cover the time and spatial coordinates, and using geometrized units with c=G=1. The $\Lambda$ term was not in the original EFEs, and have a story, but represent the dark energy. The G term on the left of the EFE is the Einstein tensor (composed of spacetime curvature components), the right hand side is the stress energy tensor.
The real question I think you are trying to get to is how is this so? What causes stress energy (in your terms, energy momentum, just somewhat different) to affect the curvature of spacetime?
There are two partial answers. One is what led to the EFE in General Relativity. It's a good question, and you are right that it was not fully answered in the other related question.
The answer was that it is due to the Einstein Field Equations (EFE) for General Relativity. That is in @QMechanics reference to How energy curves spacetime?
And that is true as far as it goes. The EFE say that spacetime is a dynamic entity, and it's curvature is determined by the matter-energy in that spacetime. The matter-energy is defined by a stress energy tensor which includes anything with momentum, anything with stress, and anything with energy. Since by E = $mc^2$ mass has an equivalence to energy, it contributes to the energy term. So the EFEs are a covariant set of equations that relate,, in any coordinate systems, the Einstein tensor to the stress energy tensor as $G_{\mu\nu} = 8\pi T_{\mu\nu}$, where the indices cover the time and spatial coordinates, using geometrized units c= G (the gravitational constant) = 1. The first term in the EFE is the Einstein tensor $G_{\mu\nu}, the next term is the dark energy term, and the right side the stress energy tensor. The dark energy terms has had its history, but it is now measured to be nonzero.
This formulation of the EFE in General Relativity (GR) has been confirmed through many measurements/observations, in realms where quantum theory is not relevant. It also has a theoretical underpinning in the equivalence principle that says that a uniform gravitational field is no different than an accelerated coordinate frame in spacetime, leading to the idea that gravity is a property of spacetime, and test particles follow geodesics in that spacetime. The equation needs to be either scalar or a tensor equation because gravity is attractive (mostly, ignore dark energy at first), and scalar equations or scalar-tensor equations were not able to reproduce observation in the solar system (nowadays there are other variations of scalar-tensor equations that have not been totally ruled out). So that led to tensor equations. The forms were dictated by having the equations depend on up to second derivatives of the metric of spacetime (since a constant term or first derivative terms could always be set to zero at a point, thus guaranteeing a local inertial reference frame), and there were not too many alternatives then. The equations also needed to reduce to the Newton's gravitation equation for a weak enough field. So there were few alternatives. There remains some efforts to find if modified versions of the EFE that predict the same observations can work, that's called Modified Gravity. So far none have predicted successfully anything observed, though they are trying because of dark energy and dark matter.
So that's the reason GR is used, and serves as an effective theory. But it still does not answer how does matter-energy create spacetime curvature. It is a classical field theory, much like Maxwell's equations were. To find a 'cause', such as a quantum theory of how it happens, we still don't know.
That would be a second tentative answer. Because it is a tensor equation with two indices, in the linearized form it can be written as a quantum field equation with spin 2 particles, the gravitons. Those would be the carriers of the 'force'. And we predict, to first order, interaction effects with other elementary particles. But they are so small as to be not measurable for now. Worse, when we try to do a quantum field theory with the full nonlinear EFEs, it turns out it is a non-renormalizable theory, partly because the graviton interacts with anything (anything with matter-energy, i.e. anything).
So we have not figured out a quantum theory of gravity. We don't know the true answer to your question. It's a current research topic.
The current hot issues/treatments include string and M theory, loop quantum gravity, and the Holographic principle that says that a gravity theory in D dimensions is equivalent to a conformal quantum field theory on its border or horizon, in D-1 dimensions.
There are a lot of unresolved issues before we can understand how to answer your question. See https://en.m.wikipedia.org/wiki/Quantum_gravity
