It is an exercise of Conway, let $p\in(0,\infty)$, if $\int_{D(0,1)} |f|^p dA<\infty$. Then classify type of singularity.
I have a very rough idea but I do not know how to write it rigorously. Assume $f$ is meromorphic, say it has a pole of order $m$. Then $mp<2$ due to integrability.
On the other hand, I think it would not be an essential sigularity. However, I do not know how to get rid of the essential singularity possibiltiy. I was thinking by Casorati Weierstrass theorem, for all $M$ there exists neighbourhood $N_M$ of size $\delta_M$ s.t. $f(z)>M$ for all $z\in N_M$. However, the size of $\delta_M$, I don't know how to control. If it goes to zero, then I cannot draw a contradiction.
Thank you!