For α, β > 0 and for a locally integrable function (or, more generally, a distribution) ψ on (0, ∞), we study the integral operators script G sign_ψ^α,β on L ² (ℝ₊) defined by (script G sign_ψ ^α,β f)(x) = ∫_ℝ+ ψ(x ^α + y^β)f(y) dy. We describe the bounded and compact operators script G sign_ψ^α,β and the operators script G sign_ψ^α,β of Schatten-von Neumann class S_p. The main results of the paper are given in Section 5, where we study continuity properties of the averaging projection script Q sign_α,β onto the operators of the form script G sign_ψ^α,β. In particular, we show that if α ≤ β and β > 1, then script G sign _ψ^α,β is bounded on S_p if and only if 2β(β + 1)^-1 < p < 2β(β - 1) ^-1.