We introduced the $q$-Picard, the $q$-Picard-Cauchy, the $q$-Gauss-Weierstrass, and the $q$-truncated Picard singular integrals. Using the last three mentioned integrals, the orders of approximation for functions from a generalized Hölder space were determined, both in the $L^p$-norm and in the generalized Hölder-norm.