As you've stated it, this generalizes trivially:
$$
P(X \ge \mathbb{E}[X] + (\mathbb{E} \vert X-\mathbb{E}[X]\vert^p)^{1/p} \ge 0) > 0
$$
for any $p>0$.  But assuming I'm right that you're interested in the probabilistic method for existence proofs (which is hard to tell since you didn't give any context), it's not really the right question to ask.  The first and second moment methods have names at all because they're frequently useful and easy to carry out.  I don't know offhand of applications where you can't use those but can use a higher $p$th moment method.  Instead one has to turn to more sophisticated tools.  Check out "The Probabilistic Method" by Alon and Spencer for a taste of lots of those.