This question is already asked here MSE, and there is an answer based on some conjecture(probably still open). I am posting the same question for a counterexample(if any, not based on such conjecture).

Problem: Let $X$ be a connected CW-complex, and $Y$ be a connected topological space. Suppose $p: X\to Y$ is a covering map. Does there exist a CW-structure on $Y$? More generally, is $Y$ homotopically equivalent to a CW-complex?

Note that the other direction is clear: Any covering of a connected CW-complex can always be given a CW-structure, lifting the characteristics map of cells of base space such that the covering map is a cellular map.

I believe that the answer to the above problem is no, but I have no counterexample.

$\bullet$ Notice that $Y$ is locally path-connected as the covering map is a local homeomorphism, hence $Y$ is path-connected also. So, we can not consider spaces $\{0\}\cup\left\{\frac{1}{n}:n\in\Bbb N\right\}$ or Topologist Sine Curve as $Y$. Notice that both $\{0\}\cup\left\{\frac{1}{n}:n\in\Bbb N\right\}$ or Topologist Sine curve are not homotopically equivalent to a CW-complex.

$\bullet$ Similarly, we can not consider the Hawaiian Earring(this is not semi-locally simply connected) as $Y$: The connected CW-complex $X$ has the universal cover so that $X$ is semi-locally simply connected, but the property "semi-locally simply connected" is preserved under a local homeomorphism.

So, I am run out of examples. Any help will be appreciated. Thanks in advance.