The authors study the approximation order by two kinds of Kantorovich type modifications from [T. Hermann and P. Vertesi, Acta Math. Acad. Sci. Hung. 37, 1-9 (1981; Zbl 0481.41019)] in \(L_w^p\) spaces with \(w(x)=\sqrt{1-x^2}.\) Namely insted of function values in nodes \(f(x_k)\) they take either weighted integral means \[ \frac{2n}{\pi }\int_{x_{k+1}}^{x_k} f(x)\frac{dx}{\sqrt{1-x^2}} \] or \[ c_{nks}^{-1}\int_{-1}^1 m_{nks}(t)f(t)\frac{dt}{\sqrt{1-t^2}}, \] where \(m_{nks}(t)\) are the corresponding fundamental functions, \(c_{nks}\) is a norming coefficient and \(x_k=x_{kn}=\cos \frac{2k-1}{2n} \pi,\) \(k=1,\ldots ,n.\)