In chapter 2 (or 1 depending on your edition) of their text Foundations of Stable Homotopy Theory, Barnes and Roitzheim have a certain Proposition 2.1.9 (or 1.1.9) saying that if $A\to X$ is an h-cofibration which is a $k$-equivalence between $(k-1)$-connected spaces, the quotient $(X,A)\to(X/A,*)$ induces isomorphisms on low-dimension homotopy groups ($n<2k$) and an epimorphism on a certain homotopy group ($n=2k$). This part of the result is standard enough. However, they then add that the result holds in $Top_*$, the category of pointed spaces, provided both $A$ and $X$ are well-pointed. It's unclear to me exactly what this means, since as far as I can tell the statement is the same for pointed spaces (the h-cofibrations are the same, k-equivalences and k-connectedness are the same, and quotients are the same).
The only interpretation I could think of is that if $A$ and $X$ are well-pointed, then the span diagram defining the pushout (for the quotient) is half-cofibrant over a left proper model category, so $X/A$ is homotopy equivalent to the space obtained by adjoining the reduced mapping cone. However, while this allows a construction similar to the unpointed case, I don't see why this is necessary, as the result appears to carry over directly.