= Solution
The count $N_t=M(0,t]$ is a rate-$\lambda$ <Poisson process>. Over a time interval $(s,t]$, the increment
$$
X_t-X_s=\sum_{n=N_s+1}^{N_t}g(Y_n)
$$
depends only on the Poisson points and marks in that interval. Disjoint intervals give independent increments, and the distribution depends only on $t-s$. The paths are càdlàg step functions, $X_0=0$, and
$$
\mathbb P(X_{t+h}\ne X_t)\leq\mathbb P(N_{t+h}-N_t\geq1)
=1-e^{-\lambda h}\longrightarrow0.
$$
Thus $X$ is stochastically continuous and
$$
\boxed{(X_t)_{t\geq0}\text{ is a compound Poisson process, hence a Lévy process}.}
$$
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