  We consider
  the Hilbert-type operator defined by
  $$
  H_{\omega}(f)(z)=\int_0^1 f(t)\left(\frac{1}{z}\int_0^z
B^{\omega}_t(u)\,du\right)\,\omega(t)dt,$$
  where $\{B^{\omega}_\zeta\}_{\zeta\in\mathbb{D}}$ are the reproducing kernels
of the Bergman space $A^2_\omega$ induced by a radial weight $\omega$ in the
unit disc $\mathbb{D}$. We prove that $H_{\omega}$ is bounded on the Hardy
space $H^p$, $1<p<\infty$, if and only if \begin{equation} \label{abs1}
\sup_{0\le r<1} \frac{\widehat{\omega}(r)}{\widehat{\omega}\left(
\frac{1+r}{2}\right)}<\infty, \tag{\dag} \end{equation}
  and
  \begin{equation*}
  \sup\limits_{0<r<1}\left(\int_0^r \frac{1}{\widehat{\omega}(t)^p}
dt\right)^{\frac{1}{p}}
  \left(\int_r^1
\left(\frac{\widehat{\omega}(t)}{1-t}\right)^{p'}\,dt\right)^{\frac{1}{p'}}
<\infty,
  \end{equation*}
  where $\widehat{\omega}(r)=\int_r^1 \omega(s)\,ds$.
  We also prove that $H_\omega: H^1\to H^1$ is bounded if and only if
\eqref{abs1} holds and
  $$ \sup\limits_{r \in [0,1)} \frac{\widehat{\omega}(r)}{1-r} \left(\int_0^r
\frac{ds}{\widehat{\omega}(s)}\right)<\infty.$$
  As for the case $p=\infty$, $H_\omega$ is bounded from $H^\infty$ to $BMOA$,
or to the Bloch space, if and only if \eqref{abs1} holds.
  In addition, we prove that there does not exist radial weights $\omega$ such
that $H_{\omega}: H^p \to H^p $, $1\le p<\infty$, is compact and we consider
the action of $H_{\omega}$ on some spaces of analytic functions closely related
to Hardy spaces.
