Assuming that $M$ and $N$ are Riemannian manifolds, the space $L^p(M,N)$ consists of measurable mappings $f:M\to N$ such that $x\to d(y_0,f(x))$ belongs to $L^p(M)$. There is no problem with this definition if the measure of $M$ is finite and a small problem if the measure of $M$ is infinite. Indeed, in the later case the constant mapping $f(x)=y_0$ belongs to $L^p$ because $d(y_0,f)\equiv 0$, but if we change the point $y_0$ to $y_1\neq y_0$, then $d(y_1,f)=d(y_1,y_0)\neq 0$ is not in $L^p$.

Actually, one can define $L^p(X,Y)$
mappings from a measure space $X$ into a separable metric space as follows:
There is an isometric embedding of $Y$ into $\ell^\infty$ and then we define:
$$
L^p(X,Y)=\{f\in L^p(X,\ell^\infty):\, f(x)\in Y \text{ a.e.}\}.
$$
Here the the $L^p$ integral of $\ell^\infty$ valued mappings is understood as the Bochner integral.

For more details see for example:

J. Heinonen, P. Koskela, N. Shanmugalingam, J. T. Tyson, *Sobolev spaces on metric measure spaces. An approach based on upper gradients.* New Mathematical Monographs, 27. Cambridge University Press, Cambridge, 2015.