The authors propose and study a class of methods for the approximation of solutions to nonlinear equations with smooth operators in a Banach space, when the operators are approximately given and their derivatives are not regular. The construction of the presented methods deals with the operator differential equation obtained by linearisation of the original equation by using the Newton-Kantorovich scheme and various ways of regularisation of it. When the initial discrepancy possesses a sourcewise representation, the authors establish estimates for the approximation errors.