Assume a Markov chain on a measurable state space $(E,\Sigma)$ is given, denoted by $(X_n)_{n\in \mathbb{N}}$ with Markov kernel $p$ and stationary measure $\mu$.
In this case, we have
$$ \mathbb{P}[X_0 \in A , \, X_1 \in B] = \int_A p(x,B) \, \mu(dx). $$
Is there a similar interpretation for the term
$$ \int_A \big(p(x,B)\big)^2 \mu(dx) ? $$
I could figure out an interpretation when $p$ is reversible i.e.
$$ \int_A p(x,B) \, \mu(dx) = \int_B p(x,A) \, \mu(dx) .$$
First by the reversible property we have $$ \int_E f(x) \int_E g(y)\, p(x,dy) \, \mu(dx) = \int_E g(x) \int_E f(y) \, p(x,dy) \, \mu(dx).$$ For $f(x) = p(x,A) 1_B(x)$ and for $g(x) = 1_A(x)$ we have
$$ \begin{align} \int_B p(x,A) \cdot p(x,A) \, \mu(dx) &= \int_E f(x) \int_E g(y) \, p(x,dy) \mu(dx) \\ &= \int_E g(x) \int_E f(y) \, p(x,dy) \, \mu(dx) \\ &= \int_A\int_B p(y,A) \, p(x,dy) \, \mu(dx) \\ &= \mathbb{P}[X_0 \in A ,X_1 \in B , X_2 \in A] \end{align}$$