Let \[ (S_{2}f)(t) := \frac{1}{\pi i}\int\limits_{0}^{\infty}\left(\frac{t}{\tau}\right)^{1/2 -1/p}\frac{f(\tau)}{\tau -t}d\tau, \quad t\in {\mathbb R}_{+}, \] be a weighted Cauchy singular integral operator, \(P^{\pm} = (I\pm S_{2})/2\), and \(U_{\alpha}:= (\alpha')^{1/p}(f\circ\alpha)\) is an isometric shift operator on \(L^{p}({\mathbb R}_{+})\) generated by an orientation-preserving diffeomorphism \(\alpha\) of \({\mathbb R}_{+}\) onto itself with the only fixed points \(0\) and \(\infty\). Let the diffeomorphism \(\beta\) satify the same assumptions. The authors study the operator \((I-cU_{\alpha}P^{+} +(I-dU_{\beta})P^{-}\) which is Fredholm on \(L^{p}({\mathbb R}_{+})\) and its index is equal to zero if \(\alpha'\), \(\beta'\), \(c\) and \(d\) are continuous on \({\mathbb R}_{+}\), slowly oscillating at \(0\) and \(\infty\), and \(\lim\limits_{t\to s}|c(t)|<1\), \(\lim\limits_{t\to s}|d(t)|<1\) for \(s=0\) and \(s=\infty\).