I don't know anything useful to say geometrically about the elliptic curves appearing in the divisor $\psi^{-1}(P)$. For instance, I don't know how to describe which $j$-invariants of elliptic curves arise in this way or even how many distinct $j$-invariants arise. Arithmetically speaking, one gets a $\mathbb{Q}$-rational divisor on $X_0(N)$ of degree equal to the modular degree (assuming $\psi$ is unramified above $P$), which is maybe the most I can say.

It works better if you go the other way: namely, choosing specific points or divisors on $X_0(N)$ and pushing them forward to $E$ yields interesting information. Two important examples:

1. The image of a cusp on $X_0(N)(\mathbb{Q})$ is a torsion point on $E(\mathbb{Q})$ (Manin-Drinfeld).

2. Starting with a point $(E,C)$ on $X_0(N)$ such that $E$ and $E/C$ both have complex multiplication by the same order in an imaginary quadratic field yields a Heegner point on $E$. A priori this point is defined over a ring class field of some imaginary quadratic field $K$, but one can take the sum of the Galois conjugates to a get a $K$-rational point. This construction has been more useful for the arithmetic of $E$ than anything else in the last $40$ years: for instance, a Heegner point is non-torsion iff $E(\mathbb{Q})$ has rank one. C.f. work of Gross-Zagier, Kolyvagin and many others.

And a remark:

3. I recall from an old paper of Birch that after mentioning Heegner points, he wonders what would happen if one took the image of a Weierstrass point on $X_0(N)$. To the best of my knowledge, this has not been explored. Although more natural from the perspective of algebraic geometry than Heegner points, much less is known about the arithmetic geometry of the Weierstrass divisor on a modular curve, for instance, the precise number field over which each of the Weierstrass points is defined.

I don't know anything useful to say geometrically about the elliptic curves appearing in the divisor $\psi^{-1}(P)$. For instance, I don't know how to describe which $j$-invariants of elliptic curves arise in this way or even how many distinct $j$-invariants arise. Arithmetically speaking, one gets a $\mathbb{Q}$-rational divisor on $X_0(N)$ of degree equal to the modular degree (assuming $\psi$ is unramified above $P$), which is maybe the most I can say.

It works better if you go the other way: namely, choosing specific points or divisors on $X_0(N)$ and pushing them forward to $E$ yields interesting information. Two important examples:

1. The image of a cusp on $X_0(N)(\mathbb{Q})$ is a torsion point on $E(\mathbb{Q})$ (Manin-Drinfeld).

2. Starting with a point $(E,C)$ on $X_0(N)$ such that $E$ and $E/C$ both have complex multiplication by the same order in an imaginary quadratic field yields a Heegner point on $E$. A priori this point is defined over a ring class field of some imaginary quadratic field $K$, but one can take the sum of the Galois conjugates to a get a $K$-rational point. This construction has been more useful for the arithmetic of $E$ than anything else in the last $40$ years: for instance, there is a nontorsion Heegner trace iff $E(K)$ has rank one. C.f. work of Gross-Zagier, Kolyvagin and many others.

And a remark:

3. I recall from an old paper of Birch that after mentioning Heegner points, he wonders what would happen if one took the image of a Weierstrass point on $X_0(N)$. To the best of my knowledge, this has not been explored. Although more natural from the perspective of algebraic geometry than Heegner points, much less is known about the arithmetic geometry of the Weierstrass divisor on a modular curve, for instance, the precise number field over which each of the Weierstrass points is defined.