Is this clique algorithm in polynomial time correct or might it have another time complexity?
Your analysis of the time complexity is wrong. Specifically this statement:

In every loop we have less n s-clique where every s-clique might have maximum s(n−1) adjacent nodes to look at.

In fact you can have up to $\binom{n}{s}$ $s$-cliques.
You approach is essentially enumerating all cliques of size up to $k-1$. So you cannot hope to have a running time better than $\Omega\left( \sum_{s=1}^{k-1} \binom{n}{s} \right)$.