Let me not Poincare dualise, and work in cohomology.
Let me generalise the setting you have described, and consider the universal surface bundle $\pi : E \to M_g^1$ over the moduli space of surfaces with one marked point, and the section $s : M_g^1 \to E$ which selects this marked point. Let $T_\pi E \to E$ be the vertical tangent bundle, with Euler class $e \in H^2(E)$. The fibrewise Poincare dual of $s$ defines a class $\nu \in H^2(E)$ which restricts to a generator of the top cohomology of each fibre $\Sigma_g$. This gives a preferred splitting of the restriction map $i^* : H^2(E) \to H^2(\Sigma_g)$. On the other hand the map $\pi^* : H^*(M_g^1) \to H^*(E)$ has a preferred splitting given by $s^*$. As $s^* \nu = s^*e \neq 0$ these splitting are not quite compatible, but $s^*(-) - s^*(e) \cdot \pi_!(-)$ gives another splitting which annihilated $\nu$. These splittings can be used to show that the Serre spectral sequence for $\pi : E \to M_g^1$ collapses, and they give a canonical decomposition
\begin{align*}
H^2(E) &= \mathbb{Z}\{\nu\} \oplus H^1(M_g^1 ; \mathcal{H}^1(\Sigma)) \oplus H^2(M_g^1)\\
x &= \nu \cdot \pi_!(x) \oplus ? \oplus (s^*(x) - s^*(e) \cdot \pi_!(x))
\end{align*}
Under this decomposition the Euler class decomposes as
$$e = (2-2g)\nu \oplus \xi \oplus s^*(e)\cdot (1-(2-2g))$$
for a twisted cohomology class $\xi \in H^1(M_g^1 ; \mathcal{H}^1(\Sigma))$. As $M_g^1$ is an Eilenberg--Mac Lane space for the mapping class group $\Gamma_g^1$, such a twisted cohomology class is represented by a crossed homomorphism
$$\Xi : \Gamma_g^1 \longrightarrow H^1(\Sigma_g)$$
(well defined up to principal crossed homomorphisms), with respect to the natural action of $\Gamma_g^1$ on $H^1(\Sigma_g)$.
To start addressing your actual question: $\xi$ is nontrivial. It is an example of a twisted Miller--Morita--Mumford class as originally studied by Kawazumi and Morita. In the notation of
N. Kawazumi, A generalization of the Morita-Mumford classes to extended mapping class groups for surfaces. Invent. Math. 131 (1998), no. 1, 137–149.
N. Kawazumi and S. Morita, The primary approximation to the cohomology of the moduli space of curves and cocycles for the stable characteristic classes, Math. Res. Lett. 3 (1996)
it is called $m_{1,1}$ (maybe up to a sign). In the notation of
A. Kupers, and O. Randal-Williams, On the cohomology of Torelli groups. Forum Math. Pi 8 (2020), e7
it is called $\kappa_{\epsilon e}$ (maybe up to a sign). Restricted to the Torelli subgroup $Tor_g^1 \leq \Gamma_g^1$ it gives a homomorphism
$$\Xi\vert_{Tor_g^1} : Tor_g^1 \longrightarrow H^1(\Sigma_g).$$
The rational abelianisation of $Tor_g^1$ is $\Lambda^3 H^1(\Sigma_g;\mathbb{Q})$, by a theorem of Johnson. This splits into irreducible $Sp_{2g}(\mathbb{Z})$-representations as $H^1(\Sigma_g;\mathbb{Q}) \oplus U$ and $\Xi\vert_{Tor_g^1}$ is is precisely detecting the part of the abelianisation map corresponding to the first summand. In particular the image of $\Xi$ is a full lattice inside $H^1(\Sigma_g)$. (I suspect it is $2 \cdot H^1(\Sigma_g)$, but cannot immediately remember why.)
Returning to your original question this means the following. For any $x \in H_1(\Sigma_g)$ there is a $N \neq 0$ and a pointed diffeomorphism $h : \Sigma_g \to \Sigma_g$, which may be taken to be in the Torelli group, such that
$$PD(e) = (2-2g)[S_p] + [N \cdot x] \in H_1(M) = \mathbb{Z}\{[S_p]\} \oplus H_1(\Sigma_g)/(h_* - Id).$$
