Polar harmonic Maaß forms and holomorphic projection

Recently, Mertens, Ono, and the third author studied mock modular analogues of Eisenstein series. Their coefficients are given by small divisor functions, have shadows classical Shimura theta functions. Here, we construct a class functions $\sigma^{\text{sm}}_{2,\chi}$ prove that these generate holomorphic part polar harmonic (weak) Maa{\ss} forms weight $\frac{3}{2}$. To this end, essentially compute projection mixed in terms Jacobi polynomials, but without assuming structure such forms. Instead, impose translation invariance suitable growth conditions on Fourier coefficients. Specializing to certain choice characters, obtain an identitiy between $\sigma^{\text{sm}}_{2,\ \text{Id}}$ Hurwitz numbers, ask for more identities. Moreover, $p$-adic congruences our when $p$ is odd prime. If $\chi$ non-trivial rewrite generating function as linear combination Appell-Lerch sums their first two normalized derivatives. Lastly, offer connection construction meromorphic index $-1$ false