Understanding a lowpass - comb filter implementation
Your diagram looks correct. Let's call the transfer function in the feedback loop $G(z)$. Consider the signal $w[n]$ at the input to the delay line. Its $\mathcal{Z}$-transform satisfies

$$W(z)=X(z)+G(z)Y(z)\tag{1}$$

where $X(z)$ and $Y(z)$ are the $\mathcal{Z}$-transforms of the input and output sequences, respectively. The output is just a delayed version of $w[n]$, i.e.,

$$Y(z)=W(z)z^{-M}\tag{2}$$

From $(1)$ and $(2)$, the transfer function of the complete system can be derived as

$$H(z)=\frac{z^{-M}}{1-G(z)z^{-M}}\tag{3}$$

Now it remains to find $G(z)$. It is quite straightforward to show that

$$G(z)=\frac{d_2f}{1-d_1z^{-1}}\tag{4}$$

Combining $(3)$ and $(4)$ gives the desired result with $d_1=d$ and assuming that $d_2=1-d_1$.