This sequent is provable.

\begin{logicproof}{2}
  (p \land q) \imp (p \land r) & \prem\\
  q & \prem\\
  \begin{subproof}
    p & \assum \\
    p \land q & $\andi 3, 2$\\
    p \land r & $\impe 4, 1$\\
    r & $\ande{2} 6$
  \end{subproof}
  p \imp r & $\impi 3-6$\\
  \begin{subproof}
    \neg(\neg p \lor r) & \assum\\
    \begin{subproof}
      \neg p & \assum\\
      \neg p \lor r & $\ori 9$\\
      \bot & $\nege 10, 8$
    \end{subproof}
    p & $\PBC 9-11 $\\
    r & $\impe 12, 7$\\
    \neg p \lor r & $\ori{2} 13$\\
    \bot & $\nege 14, 8$
  \end{subproof}
  \neg p \lor r & $\PBC 8-15$
\end{logicproof}