IMO, the fact there are some technical details left out from the homomorphism verification is secondary to a more important problem: keeping ideals of $R$ and $R/\langle x\rangle$ separate, especially when using $\langle\rangle$ to denote ideals generated by elements in both of these rings.
If you overcome this first problem I might check you on the issue of the homomorphism too, but that would be a simpler issue :) Actually, your line of reasoning contains all the seeds of a reasonable answer (modulo the way you've expressed them.)
When you write "$(R/\langle x\rangle)/\langle y\rangle$" where $\langle x\rangle$ and $\langle y\rangle$ are ideals of $R$," we immediately need to point out that anything written under "$(R/\langle x\rangle)/-$" is a subset of $R/\langle x\rangle$, not of $R$. So the only way this is meaningful is if by $\langle y\rangle$, you meant the ideal generated in $R/\langle x\rangle$ rather than in $R$. (In the original problem, you may as well just have said $y\in R$ rather than $\langle y \rangle \lhd R$.)
For clarity, you'd probably rather write $\langle y +\langle x\rangle \rangle$ rather than $\langle y\rangle$ to sidestep this issue. We will see, after all, that you are really interested in the ideal of $R/\langle x\rangle$ generated by $y+\langle x\rangle$.
With this in mind, let's look at your last line:
$r + \langle x,y \rangle \in \langle x,y \rangle \implies r \in \langle x,y \rangle \implies r+\langle x \rangle=\langle x,y \rangle \implies \ker\Psi=\langle x,y \rangle$
The first half is fine, but the last half uses some fuzzy bookkeeping with cosets that leads to a confused conclusion.
Let's start from $r\in \langle x,y\rangle$ and try something naive. We know that $r=\sum a_ixb_i +\sum c_iyd_i$ since it's in $\langle x ,y\rangle$. But then $r+\langle x\rangle$ would be $$r+\langle x\rangle=\sum a_ixb_i +\sum c_iyd_i +\langle x\rangle=\sum c_iyd_i +\langle x\rangle\in \langle y +\langle x\rangle\rangle$$.
So we should conclude that $\langle y +\langle x\rangle\rangle$ is the kernel of $\Psi$ rather than $\langle x,y \rangle$. Notice that this is a subset of $R/\langle x\rangle$, as compared to $\langle y\rangle$ which you initially emphasized to be an ideal of $R$.)
What's that look like in your original expression? As you were angling for, the first isomorphism theorem says:
$(R/\langle x\rangle)/\langle y +\langle x\rangle\rangle\cong R/\langle x,y\rangle$
So, you were headed for the right idea, but the slightly ambiguous use of notation was making you unsure.