Person A has a source $W$ with min-entropy($W$) = $k$. He also has an extra piece of information about the random source, denoted with $y$, such that min-entropy($W|y$) = $k/3$.

The adversary doesn't know $y$.

Can person A use a (strong) randomness extractor on the source $W$, where the extractor requires the min-entropy to be $k$?

For person A the output would not be close to uniform since the min-entropy given $y$ is actually $k/3$. However, for the adversary the min-entropy of $W$ is $k$. So is my reasoning correct that for the adversary the output is still close to uniform (i.e. $\epsilon$-close)?