Is this language Recursively Enumerable or Not RE?
$L = \{\text{&lt;M, k&gt;| M is a Turing Machine and } |w \in L(M) :  w \in a^*b^*|  
 \geq k \}$

Now we want to find whether $L$ is $RE$ or not.

Yes, indeed your interpretation of language $L$ is wrong.

$L$ is a language of strings of form $\langle M, k \rangle$ where $M$ is turing machine which accepts atleast $k$ strings of form $a^*b^*$.

So, now it's easy to see that $L$ is indeed $RE$. For that we can run $M$ on all strings of form $a^*b^*$ by using dove tailing approach. If $\langle M, k \rangle \in L$ then eventually $M$ will accept $k$ strings of form $a^*b^*$. And at that point we will accept $\langle M, k \rangle$.

So as per your question answer is that $L$ is $RE$.