Your question is equivalent to the following:

*Given a cellular inclusion $i : X\to Y$, when is there a retraction $r:Y \to X$?*

(Being a retraction means that $r\circ i: X\to X$ is the identity.)

The answer is usually phrased in terms of obstruction theory.

For simplicity, let's assume that $Y$ is a finite complex obtained from $X$ by attaching a single $j$-cell, i.e., $Y = X \cup_f D^j$, where $f: S^{j-1} \to X$ is the attaching map. Assume also that $X$ is a based space an $f$ is a based map.

In this case, it is easy to check that the desired retraction $r: Y \to X$ exists
if and only if the homotopy class $[f] \in \pi_{j-1}(X)$ vanishes. We can think of this class as an obstruction lying in
$$
\theta \in H^j(Y,X;\pi_{j-1}(X))
$$
(the $j$-cohomology group of the pair $(Y,X)$ with coefficients in $\pi_{j-1}(X)$).

Now, in the general case, we inductively assume that a
retraction $$r_{j-1}: X_j \cup_{X_{j-1}} Y_{j-1}\to X$$ has already been specified where
$Y_{j-1}$ is $(j-1)$-skeleton of $Y$. We wish to
extend the retraction to $Y_j$. For every cell of $Y_j$ that is not lying in $X$,
we have an obstruction in $\pi_{j-1}(X)$ defined as above. If we vary the cells, we obtain an element of
$$
H^j(Y_j,Y_{j-1} \cup X_j ;\pi_{j-1}(X))
$$
whose vanishing is both necessary and sufficient to finding an extension $r_j: X \cup Y_{j} \to X$. Notice that the displayed cohomology group is the *cellular $j$-cochains* of the pair $(Y,X)$ with coefficients in $\pi_{j-1}(X)$. It turns out that the element in question is a cocycle in this cochain complex.

However, notice we made a choice: suppose we had used a different $r_{j-1}$?

Then the obstruction can change. With a little effort one can eventually see that the obstruction changes by a coboundary. So if we take into account all the choices, the cocycle is defined only up to a coboundary.

The upshot: there is a sequence of obstructions
$$
\theta_j \in H^j(Y,X;\pi_{j-1}(X))
$$
such that $\theta_j$ is defined when $\theta_{j-1}$ vanishes.
Furthermore all the obstructions vanish iff a retraction $Y\to X$ exists.