I am going through James Maynard's paper, *Small Gaps between Primes*, and have a number of questions regarding his approach. First, I am wondering why uses weights in his approach. While I generally understand the meaning of (2.1) on pg. 3:
$$
S(N,\rho)=\sum_{N\le n<2N}\Bigl(\sum_{i=1}^k\chi_{\mathbb{P}}(n+h_i)-\rho\Bigr)w_n,
$$
where
$$
w_n = \left(\sum_{d_i | n+h_i \forall i} \lambda_{d_i,\ldots,d_k}\right)^2 \mbox{ and } \lambda_{d_i,\ldots,d_k} \approx \left( \prod\mu(d_i)f(d_1, \ldots, d_k) \right),
$$
I am struggling to understand the role of these sieve weights. What is their purpose, and what do they do? How does use of such weights help Maynard's proof?

Second, I also wonder what is the purpose of $M_k$, as defined in Proposition 4.2 on p.5. Why do we need it and what does it do?

I read through the GPY paper, and also notice that they follow a similar approach to Maynard's, so looking there did not really help.

PS. I apologise if my questions are inappropriate in any way for Mathoverflow. I am an undergraduate trying hard to understand this paper, and realised that this should be a good place to seek answers.