= Solution
If $f$ had a pole at one of the finitely many candidate points, choose a <polynomial> vanishing to sufficiently high order at all the other candidates. <Hermite interpolation> lets its remaining Taylor coefficients be chosen so that the residue of $p^2f$ at the selected pole is nonzero. A small contour around that pole would then have nonzero <integral> by the <residue theorem>, contradicting the hypothesis. Hence none of the candidate points is a pole and $f$ is entire.
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