  Consider the Hartree-type equation in $\mathbb{R}^3$ with a delta potential
formally described by $$ i \partial_t \psi = - \Delta_x \psi + \alpha \delta_0
\psi - (I_\beta \ast |\psi|^p) |\psi|^{p - 2} \psi $$ where $\alpha \in
\mathbb{R}$; $0 < \beta < 3$ and we want to solve for $\psi \colon \mathbb{R}^3
\times \mathbb{R} \to \mathbb{C}$. By means of a Poho\v{z}aev identity, we show
that if $p = (3 + \beta) / 3$ and $\alpha \geq 0$, then the problem has no
ground state at any mass $\mu > 0$. We also prove that if $$ \frac{3 +
\beta}{3} < p < \min \left(
  \frac{5 + \beta}{3}, \frac{5 + 2 \beta}{4} \right), $$ which includes the
physically-relevant case $p = \beta = 2$, then the problem admits a ground
state at any mass $\mu > 0$.
