Let \(E^s= [0,1]^s\) be the \(n\)-dimensional unit cube, \(1<p <\infty\) and \(a= (a_1, \dots, a_s) \in\mathbb{N}^s\). Denote by \(L_p^{(a)} (E^s)\) the class of functions \(f:E^s \to\mathbb{C}\) such that \[ \left|{\partial^{b_1+ \cdots+ b_s} f \over \partial x_1^{b_1} \cdots \partial x_s^{b_s}} \right |_p \leq 1 \quad \text{for} \quad 0< b_1< a_1, \dots, \;0<b_s <a_s, \] where \(|\;|_p\) means the usual \(L_p\)-norm. Let \({\mathfrak S}\) be a mesh consisting of \(N\) nodes and weights, and \(R_{\mathfrak S} (L_p^{(a)} (E^s))\) be the error of the quadrature formulas on the mesh \({\mathfrak S}\) for the class \(L_p^{(a)} (E^s)\). Define \(R_p^{(a)} (N)= \inf_{|{\mathfrak S} |=N} R_{\mathfrak S} (L_p^{(a)} (E^s))\). Furthermore, we also define \(D_p^{(a)} (N) = \inf_{|{\mathfrak S} |=N} |r_{\mathfrak S}^{(a)} |_p\), where \(r_{\mathfrak S}^{(a)}\) is the \(a\)-deviation of the mesh \({\mathfrak S}\). The author proves that they satisfy the following two-sided estimate: \[ N^{-d} (\log N)^{(\ell-1)/2} \ll R_p^{(a)} (N),\;D_p^{(a)} (N) \ll N^{-d} (\log N)^{(\ell-1)/2} \] for every \(N>1\), where \(d= \min\{a_1, \dots, a_s\}\), \(\ell= \#\{a_i \mid a_i =d,\;1\leq i \leq s\}\), and the constants in \(\ll\) depend only on \(p\) and \(a\). This estimate extends or implies certain previous results, for example W. W. L. Chen [Mathematika 27, 153–170 (1980; Zbl 0456.10027)], etc..