  Let $E \subset \mathbb{C}$ be a Borel set such that
$0<\mathcal{H}^1(E)<\infty$. David and L\'eger proved that the Cauchy kernel
$1/z$ (and even its coordinate parts $\textrm{Re}\, z/|z|^2$ and $\textrm{Im}\,
z/|z|^2$, $z\in \mathbb{C}\setminus\{0\}$) has the following property $(*)$:
the $L^2(\mathcal{H}^1\lfloor E)$-boundedness of the corresponding singular
integral operator implies the rectifiability of $E$. Recently Chousionis,
Mateu, Prat and Tolsa extended this result to any kernel of the form
$(\textrm{Re}\, z)^{2n-1}/|z|^{2n}$, $n\in \mathbb{N}$. In this paper, we prove
that the property $(*)$ is valid for operators associated to the much wider
class of kernels $(\textrm{Re}\, z)^{2N-1}/|z|^{2N}+t\cdot(\textrm{Re}\,
z)^{2n-1}/|z|^{2n}$, where $n,N$ are positive integer numbers such that $N\ge
n$, and $t\in \mathbb{R}\setminus (t_1,t_2)$ with $t_1,t_2$ depending only on
$n$ and $N$.
