During the last years, spline quasi-interpolants have been widely exploited in the numerical resolution of integral equations due to their approximation power and to their computational advantages. In particular, this work follows some recent studies that employ spline quasi-interpolating projectors, i.e. quasi-interpolants that reproduce the spline space, for the numerical resolution of Fredholm integral equations of the second kind. These studies witness the efficiency of the application of these projectors within some well-established projection methods, namely collocation and Kulkarni's methods and their iterated versions, for the numerical treatment of linear and nonlinear Fredholm integral equations of the second kind having regular and weakly singular kernels. In these works it is underlined that the propitious behaviour of quasi-interpolating projectors stands firstly in the fact that they are built using only local data and also in the substantial reduction of the dimension of the linear system related to the projection method, if compared with interpolating projectors. In the same framework, here we raise the degree of singularity of the integral operator involved in the second kind Fredholm equations, taking into account Cauchy (i.e. strongly) singular kernels, which appear in many mathematical models related to varoius fields such as electrodynamics, fluid dynamics, aerodynamics and also in the BEM approach related to some elliptic problems. Such type of integral operator forces us to define some tailor-made spline quasi-interpolating projectors or to consider their quasi$^2$ interpolating variants, here applied in the framework of the above-mentioned collocation method. Several numerical results, validating the proposed error estimates, will be presented and discussed.