Primary obstruction

Remarks

The class $\gamma(f)$ does not depend on the choice of $g$. Indeed, any two extensions of $f$ to $X_n$ are homotopic relative to $A$ on $X_{n-1}$, since $Y$ is $(n-1)$-connected. The difference of the two extensions on the $n$-cells changes $c(g)$ by a coboundary. The map $f$ extends to $X_{n+1}$ if and only if $\gamma(f) = 0$. If $X \setminus A$ has no cells of dimension above $n + 1$, then $f$ extends to $X$ if and only if $\gamma(f) = 0$.

Example: let $(X, A) = (D^{n+1}, S^n)$, $Y = S^n$ with $n \ge 1$, and $f$ the identity. Here $H^{n+1}(D^{n+1}, S^n; \pi_n(S^n)) \cong \Z$ and $\gamma(f)$ is a generator, so $f$ does not extend.

Non-example: let $Y$ be not $(n-1)$-connected, for instance $Y = S^1$ with $n = 2$. Then $f$ need not extend over $X_2$ at all, so no class in degree $n + 1$ is defined from $f$ alone.

The same construction applies to a fibration over a CW complex with $(n-1)$-connected fibre $F$. It gives the primary obstruction to a section, a class in $H^{n+1}$ of the base with coefficients in $\pi_n(F)$, local in general.