Bounded Bergman projections $$P_\omega:L_\omega^p(v)\rightarrow{L_\omega^p(v)}$$ P ω : L ω p ( v ) → L ω p ( v ) , induced by reproducing kernels admitting the representation $$\frac{1}{(1-\overline{z}\zeta)^\gamma}\int_{0}^{1} \frac{dv(r)}{1-r\overline{z}\zeta},\;\;0\leq{r}<1,$$ 1 ( 1 − z ¯ ζ ) γ ∫ 0 1 d v ( r ) 1 − r z ¯ ζ , 0 ≤ r < 1 , and the corresponding (1,1)-inequality are characterized in terms of Bekollé-Bonami-type conditions. The two-weight inequality for the maximal Bergman projection $$P_\omega^+:L_\omega^p(u)\rightarrow{L_\omega^p(v)}$$ P ω + : L ω p ( u ) → L ω p ( v ) in terms of Sawyer-testing conditions is also discussed.