This sequent is provable.

\begin{logicproof}{2}
    p \lor \lnot p & $\LEM$ \\
    \begin{subproof}
        p & \assum \\
        \lnot (p \land q) \lor p & $\ori{2} 2$
    \end{subproof}
    \begin{subproof}
        \lnot p & \assum \\
        \begin{subproof}
            p \land q & \assum \\
            p & $\ande{1} 5$ \\
            \bot & $\nege 6,4$
        \end{subproof}
        \lnot (p \land q) & $\negi 5-7$\\
        \lnot (p \land q) \lor p & $\ori{1} 8$
    \end{subproof}
    \lnot (p \land q) \lor p & $\ore 1, 2-3, 4-9$
\end{logicproof}
