My lecture notes use powers of transition kernels, but I am not sure what is meant by these powers.
Let $E$ be a Polish space equipped with the Borel $\sigma$-algebra $\mathcal{E}$, and with a partial ordening $\preceq$ on $E$.
Lemma Let $\lambda, \mu : \mathcal{E} \to [0,1]$ be probability measures s.t. $\lambda$ is stochastically dominated by $\mu$. Let $K, K^{\prime} : E\times \mathcal{E} \to [0,1]$ be transition kernels s.t. $K$ is stochastically dominated by $K^{\prime}$. Then $\lambda K^n$ is stochastically dominated by $\mu {K^{\prime}} ^{n}$ for all $n\in \mathbb{N} _0$, where $\lambda K^n$ and $\mu {K^{\prime}} ^{n}$ are to be read as probability measures on $E ^{n+1}$
What is in this context meant by $K^n$, $\lambda K^n$ and $\mu {K^{\prime}} ^{n}$?
If $K,K' \colon E \times \mathcal{E} \to [0,1]$ are two probability kernels, their product is defined by $$KK'(x,A) = \int_E K(x, dy) K'(y,A).$$ You can check that this is again a kernel. If $\lambda$ is a probability measure, $$\lambda K(A) = \int_E \lambda(dy) K(y,A)$$ defines a probability measure. In your case, $K$ is the transition kernel describing the transitions of a Markov chain $X$: $$\mathbb{P}(X_1 \in A|X_0 = x) = K(x,A);$$ $K^n$ describes the transitions of the chain over $n$ steps: $$\mathbb{P}(X_n \in A |X_0=x) = K^n(x,A);$$ and finally $\lambda K^n$ is the distribution of $X_n$ if the initial distribution $X_0 \sim \lambda$.