number theory – RSA Crypto reconstructing m by knowing (m mod p)

Let $c = m^e mod N$ be a RSA-ciphertext with $N = pq$.

$m_p = m mod p$ be known

Knowing this, how can i reconstruct $m$ in polynomial time with respect to $log N$ and $log e$ ?

I suspect it might have to do with chinese remainder theorem, but i am not entirely sure. I know we need to find m, such that

$gcd(m – m^e, N) = p$, but i dont quite see, how we can do that in polynomial time