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Ok, going off my second example from above, in that same exercise Hatcher gives a hint that solves your problem. Let $X$ be an acyclic CW-complex which isn’t contractible. Let $f: X \rightarrow *$. The mapping cone of this is $SX$, the suspension of $X$. This is acyclic and simply connected, so by exercise 8 of section 4.2 in Hatcher it's contractible. But the map $X \rightarrow *$ is not a homotopy equivalence