I am having problems either finding such a sequence or showing that one does not exist. Here is the situation.
Take $f$ to be some positive real number which is not an integer.
Recall that $\operatorname{sinc}(x)$ is $\sin(x) / x$ if $x \neq 0$, and $\operatorname{sinc}(x) := 1$.
I would like to show that no non-zero series $\{a_f(k)\}_{k=1}^\infty \subset \mathbb{R}$ exists which satisfies $$\sum_{k=1}^\infty a_f(k) \operatorname{sinc} (\pi k f) = 0.$$ (I use non-zero to simply mean that there is at least one non-zero element of the sequence.)
If it is possible to do this, then I would perhaps like to say that for $N$ a large positive integer, there is no sequence of numbers $\{a_f(k)\}_{k=1}^N \subset \mathbb{R}$ which satisfies $$\sum_{k=1}^N a_f(k) \operatorname{sinc} (\pi k f) = 0.$$ If this is also possible, then I would maybe add some conditions like $f$ being irrational so that no such sequence would exist.
As a consequence of the Poisson summation formula, for any $\alpha\in\mathbb{R}\setminus\mathbb{Z}$ we have $$ \sum_{k\geq 1}\text{sinc}(\pi k \alpha)=\frac{1}{2}\left[-1+\sum_{k\in\mathbb{Z}}\text{sinc}(\pi k \alpha)\right]=\frac{1}{2}\left[-1+\sum_{s\in\mathbb{Z}}\frac{\text{sign}(\alpha-2s)+\text{sign}(\alpha+2s)}{2\alpha}\right] $$ such that for any $\alpha\in\mathbb{R}^+\setminus\mathbb{Z}$ we get $$ \sum_{k\geq 1}\text{sinc}(\pi k\alpha) = \frac{1}{2}\left[-1+\frac{2\lfloor \alpha/2\rfloor +1}{\alpha}\right]\neq 0. $$ Similarly you may compute in explicit terms $\sum_{k\geq 1}\text{sinc}^2(\pi k \alpha)$. This disproves your claim for $$ a_f(k) = 1+ C_f\,\text{sinc}(\pi k f).$$