Restricted version of the Clique problem?
This sounds like you are asking whether or not the NP-complete clique problem can be solved in logarithmic space. Using Turing machines, one tape is read only and stores the input graph. The other tape is bounded so that there's $c \lg n$ space available for some constant $c$. The class of problems solvable in this model is known as $L$, deterministic logarithmic space. (See wikipedia or in the complexity zoo)

It is unknown whether $\mathrm{CLIQUE} \in L$, but a positive answer would imply that $P = NP$, so you (almost surely?) won't find an answer. $L \subseteq P \subseteq NP$ and $\mathrm{CLIQUE} \in L$ implies $\mathrm{CLIQUE} \in P$, which implies $P = NP$. 



Edit in case I misinterpreted the problem:

If you intend that the $k$ in $ \lg n^k = k \lg n $ is the same as the clique size $k$ (i.e. the amount of memory scales with input $k$), then there's a simple brute force algorithm: you can loop through all possible sets of $k$ nodes and check if they form a $k$-clique. A starting point for searching for better solutions might be the references of [1].



[1] Virginia Vassilevska, "Efficient algorithms for clique problems" pdf link