Linear MMSE estimator for linear signal model

Question

Let H be a complex matrix (channel matrix) of size $M \times N$, $x: Ω \to \mathbb{C}^N$ (signal), $n: Ω \to \mathbb{C}^M$ (noise) be random variables. Let $$y = Hx + n ∈ \mathbb{C}^M$$ Problem: Find the optimal solution among $\tilde{x} = Gy$, i.e. minimise $E(|Gy − x|^2)$, if $E(x)=E(n)=0, E(xx^*)=p, E(nn^*)=R$, noise n is independent of the signal x. I will not provide the solution, but we come to the conclusion that $$(HH^* + \frac{R^*}{p})G = H^*,\;\; G = (HH^* + \frac{R^*}{p})^{-1}H^*$$ assuming $(HH^* + \frac{R^*}{p})$ is invertible. But what do we do if it is not?