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It's probably more straightforward to first compute the total transfer function in terms of $H(z)$, then plug in the actual expression for $H(z)$, and from that write down the difference equation: $$Y(z)=X(z)+\alpha z^{-M}H(z)Y(z)\tag{1}$$ From $(1)$ we can derive the total transfer function: $$\frac{Y(z)}{X(z)}=\frac{1}{1-\alpha z^{-M}H(z)}\tag{2}$$ ...
It must be added to the problem that $R(\omega)$ is a real-valued, possibly bipolar function. In that case, its inverse discrete-time Fourier transform must be even: $$r[n]=r[-n]\tag{1}$$ From the given relation between $H(e^{j\omega})$ and $R(\omega)$ it is clear that $$h[n]=r[n-25]\tag{2}$$ must hold. I'm sure that you'll manage to combine $(1)$ and $(2)$ ...