If $f$ is $M$-sparse, then from Cauchy-Schwarz one has $|S(\alpha)|^2 \leq M \sum_{n \leq N} |f(n)|^2$ which gives the bound
$$ \sum_{r=1}^R |S(\alpha_r)|^2 \leq R M \sum_{n \leq N} |f(n)|^2$$
which is superior in the regime $RM \leq N$ (i.e., below the range of the Heisenberg uncertainty principle). For $RM \geq N$ one is now consistent with the uncertainty principle and one cannot hope to do better than the large sieve in general. Indeed, if $M$ divides $N$ and $f$ is the indicator function of the multiples $\{ mN/M: m=1,\dots,M\}$ of $N/M$, then $S(\alpha)=M$ for $\alpha$ a multiple of $M/N$, so if one chooses $\alpha_1,\dots,\alpha_R$ to contain the $N/M$ different multiples of $M/N$ mod $1$ (and sets $\delta$ to be anything less than or equal to $M/N$) then equality in the large sieve is essentially attained.
Note that in this example $f$ had a lot of additive structure (in particular the support was basically closed under addition). If $f$ is additively unstructured (e.g., if it is supported on a dissociated set) then one can do better by taking higher moments. For instance by applying the large sieve to $f*f$ rather than to $f$ one has
$$ \sum_{r=1}^R |S(\alpha_r)|^4 \ll (N + \delta^{-1}) \sum_{n \leq 2N} |f*f(n)|^2$$
and hence by Cauchy-Schwarz
$$ \sum_{r=1}^R |S(\alpha_r)|^2 \ll R^{1/2} (N + \delta^{-1})^{1/2} (\sum_{n \leq 2N} |f*f(n)|^2)^{1/2}.$$
If $f$ is not very additively structured, then $f*f$ will be quite spread out and this can be a superior bound to the previously mentioned bounds. Similarly if one plays with higher convolutions such as $f*f*f$. Basically one is convolving one's way out of sparsity in order to make more efficient use of the large sieve. (A similar technique is frequently employed to study large values of Dirichlet polynomials, taking advantage of the fact that the logarithms $\log n$ of the natural numbers only have a very limited amount of additive structure, thanks to the divisor bound.)
As a variant of the above strategy, if $f$ is supported in some set $E$ on which good bounds are known on exponential sums $\sum_{n \in E} e(\alpha n)$ (or weighted versions of such sums) - which is a measure of lack of additive structure in $E$ - then one can sometimes get good bounds from a duality argument (sometimes known as a "large values" argument, particularly in the context of controlling the large values of Dirichlet polynomials). See for instance
Green, Ben; Tao, Terence, Restriction theory of the Selberg sieve, with applications, J. Théor. Nombres Bordx. 18, No. 1, 147-182 (2006). ZBL1135.11049.
for an example of this (where the function $f$ is supported on something like the set of primes).
Best Answer
This is part of the theory of the "linear sieve." Chapter 8 of the book of Halberstam and Richert deals with this topic. Alternatively you could look at Iwaniec's paper On the error term in the linear sieve. The result you want can be extracted from the lower bound (1.4) on page 2 of Iwaniec's paper, together with the formula for $f(s)$ at the top of that page.