What's the problem if the plaintext is not coprime with RSA's public key N?

Question

If the plaintext (m) is to be encrypted by a RSA's publick key (N,3),

is there any problem if this plaintext m is not relatively prime with N?

(Please ignore the impact of exponent e equals 3.)

What I could think of is that they share a GCD.

Let’s say N=pq, and the GCD(m,N)=p, so m=pk.(k is an integer.)

Given e = 3:

E(m) = m^e (mod N)

     = m^3 (mod N)

     = (p.k)^3 (mod (p.q))

     = ((p.k) mod (p.q))^3 (mod (p.q))

But I can't continue.

And I'm sorry I searched in our forum but couldn't find the similar thread talking about this question.

Any suggestions are really appreciated.