Formalize pushout to map of homotopy groups

Given a finite CW-complex $$X$$ and a space $$Y$$, together with two continuous maps $$f, g\colon X \to Y$$, which agree on the $$k$$-th skeleton $$X^k$$ of $$X$$.

Given a $$(k+1)$$-cell of $$X$$ with characteristic map $$e: (D^{k+1}, S^k) \to (X^{k+1}, X^k)$$ one can ask whether $$f$$ and $$g$$ are homotopic on $$e$$. To do this, one can consider the pushout of two $$D^{k+1}$$ along $$S^k$$. Using the map $$f \circ e$$ on one side and $$g \circ e$$ on the other yields a unique map $$S^{k+1} \to Y$$, i.e. a unique element of $$\pi_{k+1}(Y)$$.

Both maps are homotopic on $$e$$ if and only if the given element is trivial in $$\pi_{k+1}(Y)$$. No magic.

I would like to formalize this process as a map $$C^{cell}_{k+1} \cong \pi_k(X^{k+1}, X^k) \to \pi_k(Y).$$

It is easy to see, that the element represented by the pushout does not depend on the homotopy class of $$e$$. This is what I have: let $$P = f(X^k) = g(X^k)$$, then $$\pi_k(X^{k+1}, X^k) \xrightarrow{f_* - g_*} \pi_k(Y, P)$$ because $$f$$ and $$g$$ agree on $$X^k$$ and the naturality of $$\delta$$, the image of $$f_* - g_*$$ is in the kernel of $$\delta$$, where $$\delta$$ is the connecting homomorphism for the long exact homotopy sequence for the pair $$(X^{k+1}, X^k)$$ resp. $$(Y, P)$$.

Hence, for the image of a cell $$e$$, there exists and element in $$\pi_k(Y)$$, which maps to the image of $$e$$. However, this element is only unique up to an element of $$\pi_k(P)$$.

At this point I am missing some tools. I would expect that the "lift" is unique, but in general I can not assume, that the inclusion of $$\pi_k(P)$$ is trivial. How do I overcome this ambiguity?