  Recently, Bruinier and Ono proved that the coefficients of certain weight
-1/2 harmonic weak Maa{\ss} forms are given as "traces" of singular moduli for
harmonic weak Maa{\ss} forms. Here, we prove that similar results hold for the
coefficients of harmonic weak Maa{\ss} forms of weight $3/2+k$, $k$ even, and
weight $1/2-k$, $k$ odd, by extending the theta lift of Bruinier-Funke and
Bruinier-Ono. Moreover, we generalize their result to include \textit{twisted}
traces of singular moduli using earlier work of the author and Ehlen. Employing
a duality result between weight $k$ and $2-k$, we are able to cover all
half-integral weights. We also show that the non-holomorphic part of the theta
lift in weight $1/2-k$, $k$ odd, is connected to the vanishing of the special
value of the $L$-function of a certain derivative of the lifted function.
