In other words:

Start with a random variable $X_0$ Bernoulli with parameter $p$, random variables $Y_n$ Bernoulli with parameter $p$, random variables $Z_n$ Bernoulli with parameter $\alpha$, and assume that all these are independent. For every $n\ge0$, define $X_{n+1}=Y_nX_n+(1-Y_n)Z_n$.

Then $X_n$ and $X_{n+k}$ are conditionally correlated if and only if $Y_i=1$ for every $i$ from $n$ to $n+k-1$. This happens with probability $\alpha^k$, hence you are done.

This is Douglas Zare's idea, but with no Poisson process.

In other words:

Start with a random variable $X_0$ Bernoulli with parameter $p$, random variables $Y_n$ Bernoulli with parameter $\alpha$, random variables $Z_n$ Bernoulli with parameter $p$, and assume that all these are independent. Define recursively the sequence $(X_n)_{n\ge0}$ by setting $X_{n+1}=Y_nX_n+(1-Y_n)Z_n$ for every $n\ge0$.

Then $X_n$ and $X_{n+k}$ are conditionally correlated if and only if $Y_i=1$ for every $i$ from $n$ to $n+k-1$. This happens with probability $\alpha^k$, hence you are done.

This is Douglas Zare's idea, but with no Poisson process.