Given a closed Riemannian manifold $(M,g)$ with non-negative Ricci curvature and $dim\geq 3$, when can we deform the metric to a positive Ricci curved one?

I know it's impossible in general due to the flat factor in the universal covering. But what about we add some topological restrictions on $M$ like simply connectedness? Are there any positive or negative results on this problem?

( Besides, are there now any examples of simply connected closed manifold with positive scalar curved metric by do not admit a positive Ricci curved metric? )

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Thanks to the answer by Robert, I may simplify the problem in the following sense,

Given a simply connected flat Einstein manifold $(M,g)$ with $\hat{A}$-genus non-vanishing, and set $(\mathbb{S}^2,h)$ be the standard unit sphere, can $(M\times \mathbb{S}^2, g+h)$ be perturbed to a Ricci-positive manifold? $\ \ $In general, what about changing $(\mathbb{S}^2,h)$ to an arbitrary closed Ricci-positive manifold?