  Bounded Bergman projections $P_\omega:L^p_\omega(v)\to L^p_\omega(v)$,
induced by reproducing kernels admitting the representation
  $$
  \frac{1}{(1-\overline{z}\zeta)^\gamma}\int_0^1\frac{d\nu(r)}{1-r\overline{z}\zeta},
  $$ and the corresponding (1,1)-inequality are characterized in terms of
Bekoll\'e-Bonami-type conditions. The two-weight inequality for the maximal
Bergman projection $P^+_\omega:L^p_\omega(u)\to L^p_\omega(v)$ in terms of
Sawyer-testing conditions is also discussed.
