If $p$ is constant, then the size of the maximum clique in the $G(n,p)$ model is almost everywhere a constant multiple of $\log n$, with the constant proportional to $\log (1/p)$. (See Bollobás, p.283 and Corollary 11.2.) Changing $p$ should therefore not affect the hardness of planting a clique with $\omega(\log n)$ vertices as long as the clique is too small for an existing algorithmic approach to work. I therefore expect that with constant $p \ne 1/2$ the hardness of Planted Clique should behave just like the $p=1/2$ case, although it is possible that the case of $p$ very close to 0 or 1 might behave differently.
In particular, for $p\ne 1/2$ the same threshold of $\Omega(n^{\alpha})$ for $\alpha = 1/2$ for the size of the planted clique applies, above which the problem becomes polynomial-time.
The value of $\alpha$ here is $1/2$ (and not some other value) because the Lovász theta function of $G(n,p)$ is almost surely between $0.5\sqrt{(1-p)/p}\sqrt{n}$ and $2\sqrt{(1-p)/p}\sqrt{n}$, by a result of Juhász.
The algorithm of Feige and Krauthgamer uses the Lovász theta function to find and certify a largest clique, so it relies on this threshold size for the planted clique.
Of course, there may be a different algorithm that does not use the Lovász theta function, and that for values of $p$ far from $1/2$ can find a planted clique with say $n^{1/3}$ vertices. As far as I can tell this is still open.
Feige and Krauthgamer also discuss when $p$ is not constant but depends on $n$, and is either close to 0 or close to 1. In these cases other approaches exist to find planted cliques, and the threshold size is different.
- Béla Bollobás, Random Graphs (2nd edition), Cambridge University Press, 2001.
- Ferenc Juhász, The asymptotic behaviour of Lovász' $\vartheta$ function for random graphs, Combinatorica 2(2) 153–155, 1982. doi:10.1007/BF02579314
- Uriel Feige and Robert Krauthgamer, Finding and certifying a large hidden clique in a semirandom graph, Random Structures & Algorithms 16(2) 195–208, 2000. doi:10.1002/(SICI)1098-2418(200003)16:2<195::AID-RSA5>3.0.CO;2-A