Let $\mathcal{M}$ be the field of meromorphic functions of one (complex) variable and $w = w(z)$ an analytic function satisfying a polynomial equation


$P(w; z) := w^n + a\_{n-1}(z) w^{n-1} + \cdots + a\_1(z) w + a\_0(z) = 0$,


where $a\_0(z), \ldots, a\_{n-1}(z)$ are in $\mathcal{M}$ (actually, is suffices to consider the case where $a\_j(z)$ is entire for $j = 0, \ldots, n-1$).


Suppose $w(z)$ has finitely many branch points.


In the (hyperelliptic) case $n=2$ it is clear that $\mathcal{M}(w) = \mathcal{M}(\sqrt{Q})$, where $Q(z)$ is a polynomial.


Is it always true that, under the above assumptions we have that $\mathcal{M}(w) = \mathcal{M}(\beta)$, where $\beta(z)$ is some algebraic function?



On finite extensions of the field of meromorphic functions
First, as you noticed, it is enough to consider the case that the equation has the form 
$$w^n+a\_{n-1}(z)w^{n-1}+\ldots+a\_0(z)=0,$$
where the coefficients are entire. Then $w$ is holomorphic on its Riemann surface, let us call this Riemann surface $S$. From your condition follows that $S$ is a compact Riemann surface with finitely many punctures. So it can be represented by an algebraic curve $K$ in $C^2$ given as a zero set of a polynomial $F(z,u)=0$. Suppose that this curve is non-singular.
Let $m=\deg\_u F$.
We have an analytic function $w$ on $K$, so it can be extended to the whole $C^2$,
So $w$ is a restriction of $K$ of an entire function $G(z,u)$.
Now on $K$ we have
$$G(z,u)=\sum\_{k,j}a\_{k,j}z^ku^j=\sum\_{j=0}^{m-1} u^j\sum\_{k,i=0}^\infty a\_{k,j+im}z^k=\sum\_{j=0}^{m-1}b\_j(z)u^j,$$
where the rearrangement of the infinite sum is legitimate because of the absolute convergence. This proves your statement as $u$ is algebraic over $C(z)$.


It may happen that every realization of $S$ in $C^2$ is singular. In this case we realize $S$ as a non-singular curve $K$ in $C^n$ (I suppose one can take $n=3$ but this is irrelevant.) Let the coordinates in $C^n$ be
$(z,u\_1,\ldots,u\_{n-1})$. Then $w$ can be represented by an entire function
$G(z,u\_1,\ldots,u\_n)$ and the restrictions on $K$ of the coordinate functions $u\_1,\ldots,u\_{n-1}$ are algebraic functions of $z$, and by the theorem on the primitive element, they are all rational functions of $z$ and some $\beta$, where $\beta$ is an algebraic function of $z$. Then the same argument works. 


