In RSA, what's the risk of revealing $p^e\bmod N$?
Does revealing $p^e\bmod N$ (in addition to the public key) allow factorization of $N$


Yes, with that extra information, $N$ is easy to factor.  A computation of $\gcd( N, p^e \bmod N )$ gives the factor $p$ (because $p$ is the only prime the two integers $N$ and $p^e \bmod N$ have in common, it is a factor of both integers, while $q$ is not a factor of $p^e \bmod N$)