Let $B$ be a finite CW complex and $p:E\to B$ a Hurewicz fibration.
I know that every topological space $E$ has the weak(!) homotopy type of a CW complex $D$ realized by a weak equivalence $f:D\to E$. (I apologize for asking two things in one question but they are so related that splitting them up seems a little exaggerated.)
Is it possible to find a CW complex $D'$, a Hurewicz fibration $p':D'\to B$ and a weak equivalence $f':D'\to E$ such that $p'=p\circ f'$?
Is it possible to find a Serre fibration $q:C\to B$ and a weak equivalence $g:C\to E$ such that $q=p\circ g$? Can I choose $C$ to be a CW complex?
Due to Henry T. Horton's great answer, the whole question reduces to the following question.
Let $E$ be a topological space. Is it possible to find a CW complex $D$ and a weak equivalence $f:D\to E$ such that $f$ is also a Serre fibration?