Studying a problem in conformal geometry, I am facing to the following interpolation problem.

Let $P$ and $Q$ two coprime polynomials. Then let $A$ and $B$ two coprime polynomials such that $$\frac{A}{B} = \left( \frac{P}{Q}\right)'.$$

We denote by $(z_i)_{0\leq i\leq n}$ some distinct complex numbers with $n=deg(A)$ and which are not some roots of $Q$. Then we consider some perturbations of the $\left( \frac{P}{Q}\right)'(z_i)$ denoted $\tilde{a}_i$ and we try find a perturbation of $P$ and $Q$ denoted $\tilde{P}$ and $\tilde{Q}$ such that

$$\left( \frac{\tilde{P}}{\tilde{Q}}\right)'(z_i)=\tilde{a}_i$$

The problem is easy to solve when $Q\equiv 1$, it is classical interpolation on $P'$ and then we integrate. But When $Q$ is not trivial we can't just perturb $A$, finding $\tilde{A}$ such that

$$\tilde{A}(z_i)= \tilde{a_i} B(z_i)$$

because when integrating $\frac{\tilde{A}}{B}$ there is no reason to get a rational fraction.

I have tried to perturb successively $P$ and $Q$(or $A$ and $B$) and to get some estimate to make a fixed point but without success? did any one get some references or ideas on that topic?

The Geometric point of view is the following, you consider a (finite) multiple conformal covering of $S^2$, then identifying $S^2$ with $\hat{\mathbb{C}}$ it is parametrized by a rational fraction. Then you can define the vector field $ \left( \frac{P}{Q}\right)'$. And the question is, if you perturb this vector field at $n+1$ points which are not poles, can you reach this perturbation by perturbing $P$ and $Q$?

Edit: The first question is of course the existence of $\tilde{P}$ and $\tilde{Q}$? But I would like that $\tilde{P}$ and $\tilde{Q}$ will be closed to $P$ and $Q$ with respect to $\sum \vert a_i -\tilde{a}_i\vert$. Closed means for me, with the same degree and closed in $\mathbb{C}_d[X]$ with $d$ the degree considered.