Let $p$ a prime and $N$ an integer such that $p\not\mid N$. I will denote with $X_0(m)$ the modular curve with respect to the congruence subgroup $\Gamma_0(m)$.
Let $f$ be a modular form with respect to $\Gamma_0(N)$ of weight $k$ and consider the $p$-depletion defined by
$f^{[p]}(\tau)=f(\tau)-a_pf(p\tau)+p^{k-1}f(p^2\tau)$,
where $a_p$ is the $p$-th coefficient of the $q$-expansion of $f$ at infinity. With the usual interpretation of a modular form of level $N$ as functions on triples $(E,t,\omega)$, seen as $\mathbb{C}$-points of $X_0(N)$, with $t$ subgroup of order $N$ of the elliptic curve $E/\mathbb{C}$ and $\omega$ a chosen differential, I would like to ''evaluate'' $f^{[p]}$ in such a point, at least when $E$ is ordinary at $p$.
A priori $f^{[p]}$ is a modular form of level $p^2N$. So one considers the natural projection $\pi:X_0(p^2N) \rightarrow X_0(N)$. My question is: does this admit a section on the ordinary locus
$s: \mathcal{A}\rightarrow X_0(p^2N)(\mathbb{C}_p)$
so that one can use this to put $f^{[p]}((E,t)):=f^{[p]}(s((E,t)))$? If so how is this section defined for example in the simplest case of $E$ with CM by $K$ with $p=\mathfrak{p}\bar{\mathfrak{p}}$ splitting in $K$? Is this the correct way of viewing this?
Yes, there is a section $X_0(N)^{\mathrm{ord}} \to X_0(Np^r)^{\mathrm{ord}}$ for any $r \ge 1$, indentifying $X_0(N)^{\mathrm{ord}}$ with the connected component of $X_0(Np^r)^{\mathrm{ord}}$ containing $\infty$.
This is actually not difficult to see. If $E$ is an elliptic curve (let's say over $\mathcal{O}_{\mathbf{C}_p}$) whose mod $p$ reduction $\overline{E}$ is non-singular and ordinary, then $E[p^n]$ has order exactly $p^{2n}$, but $\overline{E}[p^n]$ has order $p^n$ (because the multiplication-by-$p$ isogeny has separable degree $p$). So the kernel of $E[p^n] \to \overline{E}[p^n]$ has order $p^n$, and that gives you a canonical cyclic $p^n$-subgroup of $E$. With a little bit of work one can check that this construction extends over the cusps too, so it gives a section $X_0(N)^{\mathrm{ord}} \to X_0(Np^r)^{\mathrm{ord}}$.
What is much harder (but also true) is that this section will extend to a strict neighbourhood of $X_0(N)^{\mathrm{ord}}$ in $X_0(N)$ -- it is "overconvergent". This is explained in Katz's article in the Antwerp proceedings (Springer Lecture Notes #330).
For a CM elliptic curve the canonical subgroup will be $E[\mathfrak{p}^2]$ (for some choice of embedding $K \hookrightarrow \overline{\mathbf{Q}}_p$, which singles out one of the two primes above $p$).