Let $U$ be a bordered Riemann surface of genus $g$ with $n -1$ punctures and one hole (i.e., the border has one connected component). For any punctured Riemann surface $\Sigma$ of genus $g$ with $n$ punctures, does there necessarily exist an embedding (i.e. an injective holomorphic map) $\Phi: U \to \Sigma$?

The answer is no. For $g=0$, the problem is equivalent to the following: is there a univalent function in the unit disk which takes given values at finitely many given points. It is well known that the answer is no: there are many inequalities relating the points and the values. For example, Theorem 1 on p. 119 of Goluzin's book Geometric theory of functions of a complex variable, Engl. tranls. AMS, 1969 (available on the Internet). Certainly, similar situation prevails for other genera.

The intuitive explanation is this: suppose all your points (punctures) are far away from the hole. And two of them are very close together, while the third is on certain distance from these two. Then this situation (two close together and one relatively far away must persist no matter how you embed it.