  Hoffstein and Hulse recently introduced the notion of shifted convolution
Dirichlet series for pairs of modular forms $f_1$ and $f_2$. The second two
authors investigated certain special values of symmetrized sums of such
functions, numbers which are generally expected to be mysterious transcendental
numbers. They proved that the generating functions of these values in
$h$-aspect are linear combinations of mixed mock modular forms and quasimodular
forms. Here we examine the special cases when $f_1=f_2$ where, in addition,
there is a prime $p$ for which $p^2$ divides the level. We prove that the mixed
mock modular form is a linear combination of at most two weight 2 weakly
holomorphic $p$-adic modular forms.
