$D(P,r)\backslash \{P\}\xrightarrow{f}U\xrightarrow{g}\mathbb{C}$. Both $f$ and $g$ are holomorphic. If $f$ has a removable (or pole, or essential) singularity at $P$, does $g\circ f$ also have one?
I think in any case $g\circ f$ will not have singularities any more. It is just a hunch, and I cannot explain it clearly.