  By the theory of Eisenstein series, generating functions of various divisor
functions arise as modular forms. It is natural to ask whether further divisor
functions arise systematically in the theory of mock modular forms. We
establish, using the method of Zagier and Zwegers on holomorphic projection,
that this is indeed the case for certain (twisted) "small divisors" summatory
functions $\sigma_{\psi}^{\mathrm{sm}}(n)$. More precisely, in terms of the
weight 2 quasimodular Eisenstein series $E_2(\tau)$ and a generic Shimura theta
function $\theta_{\psi}(\tau)$, we show that there is a constant
$\alpha_{\psi}$ for which $$ \mathcal{E}^{+}_{\psi}(\tau):=
\alpha_{\psi}\cdot\frac{E_2(\tau)}{\theta_{\psi}(\tau)}+
\frac{1}{\theta_{\psi}(\tau)} \sum_{n=1}^\infty \sigma^{\mathrm{sm}}_\psi(n)q^n
$$ is a half integral weight (polar) mock modular form. These include
generating functions for combinatorial objects such as the Andrews
$spt$-function and the "consecutive parts" partition function. Finally, in
analogy with Serre's result that the weight $2$ Eisenstein series is a $p$-adic
modular form, we show that these forms possess canonical congruences with
modular forms.
