= Solution
Away from $p,q,r$, all three coordinate functions are holomorphic and the first is one, so they define a projective point. Near $p$, take a local coordinate $z$ vanishing there. Write $f=a/z+O(1)$ with $a\ne0$, while $g$ is holomorphic. Then
$$
[1:f:g]=[z:zf:zg]\longrightarrow[0:a:0]=[0:1:0].
$$
The scaled coordinates are holomorphic and do not all vanish, proving a holomorphic extension. Similarly $F(r)=[0:0:1]$. At $q$, write $f=b/z+O(1)$ and $g=c/z+O(1)$ with $b,c\ne0$. The same scaling gives $F(q)=[0:b:c]$, again well-defined. Hence
$$
\boxed{F:C\longrightarrow\mathbb P^2\text{ is a well-defined holomorphic map}.}
$$
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