Consider the stunted complex projective space $\mathbb{C}P^{n+2}_n:=\mathbb{C}P^{n+2}/\mathbb{C}P^{n-1}$ which is a three-cell complex of the form
$$\mathbb{C}P^{n+2}_n\simeq S^{2n}\cup_{\alpha_n} e^{2n+2}\cup_{\beta_n} e^{2n+4}.$$
I am interested in knowing the attaching maps $\alpha_n$ and $\beta_n$.
To begin we note that the cohomology $H^*\mathbb{C}P^{n+2}_n=\mathbb{Z}\{x^n,x^{n+1},x^{n+2}\}$ is free on three classes which correspond via the collapse map $\mathbb{C}P^{n+2}\rightarrow \mathbb{C}P^{n+2}_n$ with the top three powers of the canonical generator $x\in H^2\mathbb{C}P^{n+2}$. This makes it easy to calculate Steenrod squares in $H^*\mathbb{C}P^{n+2}_n$, and in particular we find that $Sq^2x^n=n\cdot x^{n+1}$, which tells us that
$$\alpha_n\simeq\begin{cases}\ast&n\equiv 0\mod 2\\\eta_{2n}&n\equiv 1\mod2\end{cases}$$
since the Hopf map $\eta_{2n}$ is detected by $Sq^2$.
It is now the even case that really interests me so I will assume from here on out that $n$ is even and also that $n>2$, so that we will be working in the stable range. Moreover I happen to only be interested in $2$-local data, and for this reason all spaces and groups will be localised at the prime $2$ in the following.
Then under these assumptions we have
$$\mathbb{C}P^{n+2}_n\simeq (S^{2n}\vee S^{2n+2})\cup_{\beta_n} e^{2n+4}$$
and $\beta_n$ identifies as an element of $\pi_{2n+3}S^{2n}\oplus\pi_{2n+3}S^{2n+2}$ since there are no Whitehead products in these dimensions. We know from the previous calculation that $\beta_n$ attaches non-trivially to $S^{2n+2}$ by the Hopf map $\eta_{2n+2}$, so recalling that $\pi_{2n+3}S^{2n}\cong\mathbb{Z}_8$ is stable and generated by the element $\nu_{2n}$ it must be that
$$\beta_n=a\cdot \nu_{2n}+\eta_{2n+2}:S^{2n+3}\rightarrow S^{2n}\vee S^{2n+2}$$
for some mod $8$ integer $a$.
Now $Sq^4$ detects $\nu_{2n}$, and with a little work we can calculate the action of $Sq^4$ on $H^*\mathbb{C}P^{n+2}_n$, finding that
$$Sq^4x^n=\begin{cases}0&n\equiv 0,1\mod 4\\ x^{2n+2}&n\equiv {2,3}\mod 4\end{cases}.$$
It follows from this that $a$ is odd when $n\equiv2\mod 4$, and this sorts out $\beta_n$ completely in these cases (we already have that $n$ is even). Therefore let us add the assumption that $n\equiv 0\mod 4$. Then there are essentially three choices for $a$. Namely $a=0,2$ or $4$.
For which even values of $n>2$ does $a=0,2,4$?
Now Adams has constructed stable secondary cohomology operations $\varphi_{0,2}$ based on the Adem relation $Sq^4Sq^1+(Sq^2Sq^1)Sq^2+Sq^1Sq^4=0$ and proved that these operations detect $2\cdot\nu$. For instance $\varphi_{0,2}(x^4)=x^6$ with no intederminacy, and this sorts out $\beta_4=2\cdot\nu+\eta$. However I am not sure of exactly how to evaluate these operations on the higher classes in $H^*\mathbb{C}P^{n+2}$. I have constructed a rudimentary Cartan formula for the operation, but am struggling to deal with the indeterminancy. Moreover, this operation will still leave undecided the cases when it vanishes, since it cannot distinguish $a=0$ and $a=4$.
I am aware of a tertiary cohomology operation capable of detecting $\eta^3=4\cdot\nu$, but I have no idea how to evaluate it on $\mathbb{C}P^{n+2}_n$ for large $n$.
I believe all this must be well known and must appear in a textbook or paper somewhere, but if it does I am not sure where. Any help tracking down reliable references would be very welcome.
This is calculated in
Specifically, in Proposition 5.2 (proved in section 9), Mosher shows in particular when $n$ is even that the attaching map of the top cell in $\mathbb{C}P^{n+2}/\mathbb{C}P^{n-1}$ is $$\eta + \lambda \nu \in \pi_{2n+3}(S^{2n+2}) \oplus \pi_{2n+3}(S^{2n})$$ where $$\lambda = \begin{cases} 0, & n \equiv 0 \pmod{8}, \\ 1, & n \equiv 2 \pmod{4}, \\ 2, & n \equiv 4 \pmod{8}. \end{cases}$$