Intersection of algebraic curves at a point with given multiplicity

Question

I don't know if this question is too basic for MO, so I put it here, but if you think I should migrate the question to MathOverflow please suggest me.

1. Let $C/k$ be a smooth curve over a perfect field, $P\in C(k)$ and $n\in \mathbb{Z}^{+}$ such that $n>1$

I want to find a $\psi\in k(C)$ such that $P\in Supp(div(\psi))$ and $\#\lbrace Supp(div(\psi)_0)\setminus \lbrace P \rbrace\rbrace\leq deg(\psi)-n$

This in other words means that $nP$ appears in $div(\psi)$ with maybe other points in the support.

For $n=2$ this $\psi$ is easy to build as a "tangent line" to $C$ at $P$.

2. If $C$ is a hyperelliptic curve and $n>2$, is it possible to find a $\psi$ such that:

   $div(\psi) = [nP -n\infty] \sim [P_1+P_2+\ldots+P_r-r\infty]$ where $r\leq genus(C)$?

I excluded $n=1$ because for $g=1$ this $\psi$ does not exist according to Riemann-Roch. Also excluded $n=2$ because we would have that $P$ is a Weierstrass point for $g>1$. Also $C$ has only one point at infinity here.

This means that $P$ is the only finite point appearing in the support of $div(\psi)$ for the left hand. For the right hand we know that we can reduce this divisor to have $r$ points in its support by RR.

In summary:

I just want to know for genus 2 if I can build the polynomial function of degree $\geq n$ such that it intersects at $P\in C$ with multiplicity $n$.

For example an application of this for a hyperelliptic curve of genus 2 with odd degree is that if I can find a $\psi\in k(C)$ such that $deg(\psi)=n+2$ and for $P\in C$ a general point, $\psi$ intersects $C$ at $P$ with multiplicity $n$, then $div(\psi)=nP+Q_1+Q_2-(n+2)\infty$, with this we have that in $Jac(C)$ and $[n]\in End(Jac(C))$ the $\psi$ polynomial will define the multiplication map by n of the prime divisor at $P\in C$, this is $[n] ([P-\infty]) = [Q_1+Q_2 - 2\infty]$.

For $n=2$ in the context of question 1 is just finding the "tangent line" to $C$ at $P$, but I would like to find a degree $\geq n$ polynomial passing "tangent" to $P$ at $C$ with multiplicity $n$ at $P$.

Maybe if I know how to build for a polynomial with $degree\geq 2$ and the case $n=2$ for $C$ a hyperelliptic curve, it would be easier to generalize it. I know that this polynomial will grow very fast , as the $[n]\in End(Jac(C))$ has degree $n^{2g}$.