The author considers the class of \(\delta\)-Calderón-Zygmund operators, which includes all classical singular integrals with kernels \(K(x,y)=\Omega(x-y)| x-y| ^n\) such that \(\Omega\in Lip_\delta\), \(\Omega(rx)=\Omega(x)\) and \(\int_{S^{n-1}}\Omega \, d\sigma=0\). If \(\omega\in A_q\), the Muckenhoupt class, and \(T\) is a \(\delta\)-Calderón-Zygmund operator such that \(T^*1\in Lip_\epsilon\), the main theorem states that \(T\) is bounded from the weighted Hardy space \(H_\omega^p(R^n)\) to the weighted local Hardy space \(h_\omega^p(R^n)\). The parameters must satisfy the conditions \(1\leq q <(n+\delta)/n\), \(q\leq (n+\epsilon)/n\), \(nq/(n+\delta)<p\leq 1\) and \(n/(n+\epsilon)\leq p\). An application to the Calderón commutator is also given.